To find the value of the function $$f(x) = 4x - 5$$ when $$x = 2$$, substitute the numerical value $$2$$ into the equation wherever the independent variable $$x$$ appears. In function notation, this step is written as $$f(2) = 4(2) - 5$$. Next, simplify the resulting expression by first performing multiplication to obtain $$8 - 5$$. Finally, subtracting gives $$3$$. Therefore, the value of the function at $$x = 2$$ is $$3$$, which is expressed concisely in function notation as $$f(2) = 3$$.

Example: Evaluating the Function $$f(x) = 4x - 5$$ at $$x = 2$$

To evaluate the quadratic function $$f(x) = 2x^2 + 3x - 1$$ for a specific value or variable, substitute that value for $$x$$ in the expression and simplify using the order of operations:

- **Evaluate $$f(3)$$:** Replace $$x$$ with $$3$$ to get $$f(3) = 2(3)^2 + 3(3) - 1$$. Simplify the exponent to get $$2(9) + 3(3) - 1$$. Multiply to obtain $$18 + 9 - 1$$. Combine the terms to find $$f(3) = 26$$.
- **Evaluate $$f(-2)$$:** Replace $$x$$ with $$-2$$ using parentheses to preserve the negative signs: $$f(-2) = 2(-2)^2 + 3(-2) - 1$$. Simplifying the exponent gives $$2(4) - 6 - 1$$. Multiply to obtain $$8 - 6 - 1$$. Simplify to find $$f(-2) = 1$$.
- **Evaluate $$f(a)$$:** Replace $$x$$ with the variable $$a$$: $$f(a) = 2(a)^2 + 3(a) - 1$$. Simplify to yield $$f(a) = 2a^2 + 3a - 1$$. The same substitution procedure applies when evaluating a function with a variable rather than a constant.

Example: Evaluating the Function $$f(x) = 2x^2 + 3x - 1$$

To evaluate the quadratic function $$f(x) = 3x^2 - 2x + 1$$ for a specific value or variable, substitute that value for $$x$$ in the expression and simplify using the order of operations:  - **Evaluate $$f(3)$$:** Replace $$x$$ with $$3$$ to get $$f(3) = 3(3)^2 - 2(3) + 1$$. Simplify the exponent to get $$3(9) - 2(3) + 1$$. Multiply to obtain $$27 - 6 + 1$$. Combine the terms to find $$f(3) = 22$$. - **Evaluate $$f(-1)$$:** Replace $$x$$ with $$-1$$ using parentheses to preserve the negative signs: $$f(-1) = 3(-1)^2 - 2(-1) + 1$$. Simplifying the exponent gives $$3(1) - (-2) + 1$$. Multiply to obtain $$3 + 2 + 1$$. Simplify to find $$f(-1) = 6$$. - **Evaluate $$f(t)$$:** Replace $$x$$ with the variable $$t$$: $$f(t) = 3(t)^2 - 2(t) + 1$$. Simplify to yield $$f(t) = 3t^2 - 2t + 1$$. This illustrates that evaluating a function with a variable produces an algebraic expression rather than a numerical value.

Example: Evaluating the Function $$f(x) = 3x^2 - 2x + 1$$

To evaluate the quadratic function $$f(x) = 2x^2 + 4x - 3$$ for a specific value or variable, substitute that value for $$x$$ in the expression and simplify using the order of operations:  - **Evaluate $$f(2)$$:** Replace $$x$$ with $$2$$ to get $$f(2) = 2(2)^2 + 4(2) - 3$$. Simplify the exponent to get $$2(4) + 4(2) - 3$$. Multiply to obtain $$8 + 8 - 3$$. Combine the terms to find $$f(2) = 13$$. - **Evaluate $$f(-3)$$:** Replace $$x$$ with $$-3$$ using parentheses to preserve the negative signs: $$f(-3) = 2(-3)^2 + 4(-3) - 3$$. Simplifying the exponent gives $$2(9) - 12 - 3$$. Multiply to obtain $$18 - 12 - 3$$. Simplify to find $$f(-3) = 3$$. - **Evaluate $$f(h)$$:** Replace $$x$$ with the variable $$h$$: $$f(h) = 2(h)^2 + 4(h) - 3$$. Simplify to yield $$f(h) = 2h^2 + 4h - 3$$. The same substitution procedure applies when evaluating a function with a variable instead of a constant.

Example: Evaluating the Function $$f(x) = 2x^2 + 4x - 3$$

To evaluate the linear function $$g(x) = 3x - 5$$ for specific algebraic expressions, substitute each expression for the independent variable $$x$$ and simplify using the order of operations:
- Evaluate $$g(h^2)$$: Replace $$x$$ with $$h^2$$ to get $$g(h^2) = 3(h^2) - 5$$. Simplify to yield $$g(h^2) = 3h^2 - 5$$.
- Evaluate $$g(x+2)$$: Replace $$x$$ with the binomial $$(x+2)$$ using parentheses to get $$g(x+2) = 3(x+2) - 5$$. Distribute the $$3$$ to obtain $$3x + 6 - 5$$. Simplify by combining the constant terms to find $$g(x+2) = 3x + 1$$.
- Evaluate $$g(x) + g(2)$$: This expression requires finding the value of $$g(2)$$ first and adding it to the expression for $$g(x)$$. Find $$g(2) = 3(2) - 5 = 6 - 5 = 1$$. Then, substitute both parts into the sum to get $$g(x) + g(2) = (3x - 5) + (1)$$. Simplify by combining the constant terms to find $$g(x) + g(2) = 3x - 4$$.

Notice that evaluating a sum, such as $$g(x+2)$$, is not necessarily equal to the sum of the evaluated function values, $$g(x) + g(2)$$.

Example: Evaluating the Function $$g(x) = 3x - 5$$

To evaluate the linear function $$g(x) = 4x - 7$$ for specific variables or algebraic expressions, substitute each expression for $$x$$ and simplify:
- Evaluate $$g(m^2)$$: Replace $$x$$ with $$m^2$$ to get $$g(m^2) = 4(m^2) - 7$$. Simplify to yield $$g(m^2) = 4m^2 - 7$$.
- Evaluate $$g(x-3)$$: Replace $$x$$ with the binomial $$(x-3)$$ using parentheses to get $$g(x-3) = 4(x-3) - 7$$. Distribute the $$4$$ to obtain $$4x - 12 - 7$$. Simplify by combining the constant terms to find $$g(x-3) = 4x - 19$$.
- Evaluate $$g(x) - g(3)$$: This expression requires finding the numerical value of $$g(3)$$ first and subtracting it from the algebraic expression for $$g(x)$$. Find $$g(3) = 4(3) - 7 = 12 - 7 = 5$$. Then, formulate the difference to get $$g(x) - g(3) = (4x - 7) - (5)$$. Simplify by combining the constant terms to find $$g(x) - g(3) = 4x - 12$$.

Example: Evaluating the Function $$g(x) = 4x - 7$$

To evaluate the linear function $$h(x) = 2x + 1$$ for specific variables or algebraic expressions, substitute each expression for $$x$$ and simplify:
- Evaluate $$h(k^2)$$: Replace $$x$$ with $$k^2$$ to get $$h(k^2) = 2(k^2) + 1$$. Simplify to yield $$h(k^2) = 2k^2 + 1$$.
- Evaluate $$h(x+1)$$: Replace $$x$$ with the binomial $$(x+1)$$ using parentheses to get $$h(x+1) = 2(x+1) + 1$$. Distribute the $$2$$ to obtain $$2x + 2 + 1$$. Simplify by combining the constant terms to find $$h(x+1) = 2x + 3$$.
- Evaluate $$h(x) + h(1)$$: This expression requires finding the numerical value of $$h(1)$$ first and adding it to the algebraic expression for $$h(x)$$. Find $$h(1) = 2(1) + 1 = 2 + 1 = 3$$. Then, substitute both parts into the sum to get $$h(x) + h(1) = (2x + 1) + (3)$$. Simplify by combining the constant terms to find $$h(x) + h(1) = 2x + 4$$.

Example: Evaluating the Function $$h(x) = 2x + 1$$

Consider an email account containing $$75$$ unread emails, growing by $$10$$ emails per day. This scenario can be modeled by the function $$N(t) = 75 + 10t$$, where the number of unread emails, $$N$$, is the dependent variable because it relies on the number of days, $$t$$, which is the independent variable. To find the number of unread emails after $$5$$ days, evaluate the function at $$t = 5$$ by substituting the value into the equation: $$N(5) = 75 + 10(5)$$. Simplifying this expression gives $$N(5) = 75 + 50$$, which evaluates to $$125$$. This result means that after $$5$$ days, there are $$125$$ unread emails in the account.

Example: Evaluating the Function $$N(t) = 75 + 10t$$ for Unread Emails

Consider an email account starting with $$100$$ unread emails and growing by $$15$$ unread emails each day. This relationship is modeled by the function $$N(t) = 100 + 15t$$, where the number of unread emails, $$N$$, is the dependent variable and the time in days, $$t$$, is the independent variable. To find the amount of unread emails after $$7$$ days, evaluate the function for $$t = 7$$ by substituting the value into the equation: $$N(7) = 100 + 15(7)$$. Simplifying this expression yields $$N(7) = 100 + 105$$, which sums to $$205$$. This indicates that after $$7$$ days, the account will have $$205$$ unread emails.

Example: Evaluating the Function $$N(t) = 100 + 15t$$ for Unread Emails

The number of unread emails in an account that initially has $$110$$ unread emails and accumulates $$25$$ more each day can be modeled by the function $$N(t) = 110 + 25t$$. Here, the number of unread emails, $$N$$, is the dependent variable, and the number of days, $$t$$, is the independent variable. To find the number of unread emails after $$14$$ days, evaluate the function at $$t = 14$$ by substitution: $$N(14) = 110 + 25(14)$$. Simplifying this expression yields $$N(14) = 110 + 350$$, which equates to $$460$$. This means there are $$460$$ unread emails in the account after a period of $$14$$ days.

Example: Evaluating the Function $$N(t) = 110 + 25t$$ for Unread Emails

To evaluate the polynomial function $$f(x) = 3x^2 + 2x - 15$$ for specific values, substitute the value for the variable $$x$$ and simplify using the order of operations:

- **Evaluate $$f(3)$$:** Substitute $$3$$ for $$x$$ to obtain $$f(3) = 3(3)^2 + 2(3) - 15$$. Simplifying the exponent gives $$3(9) + 6 - 15$$. Multiply to get $$27 + 6 - 15$$, and simplify to find $$f(3) = 18$$.
- **Evaluate $$f(-5)$$:** Substitute $$-5$$ for $$x$$ to obtain $$f(-5) = 3(-5)^2 + 2(-5) - 15$$. Simplifying the exponent gives $$3(25) - 10 - 15$$. Multiply to get $$75 - 10 - 15$$, and simplify to find $$f(-5) = 50$$.
- **Evaluate $$f(0)$$:** Substitute $$0$$ for $$x$$ to obtain $$f(0) = 3(0)^2 + 2(0) - 15$$. Simplifying the exponent gives $$3(0) + 0 - 15$$, which simplifies to $$f(0) = -15$$.

Evaluating the Polynomial Function $$f(x) = 3x^2 + 2x - 15$$

To evaluate the polynomial function $$g(x) = 5x^2 - x - 4$$ for specific values, substitute the value for the variable $$x$$ and simplify using the order of operations:

- **Evaluate $$g(-2)$$:** Substitute $$-2$$ for $$x$$ to obtain $$g(-2) = 5(-2)^2 - (-2) - 4$$. Simplifying the exponent gives $$5(4) + 2 - 4$$. Multiply to get $$20 + 2 - 4$$, and simplify to find $$g(-2) = 18$$.
- **Evaluate $$g(-1)$$:** Substitute $$-1$$ for $$x$$ to obtain $$g(-1) = 5(-1)^2 - (-1) - 4$$. Simplifying the exponent gives $$5(1) + 1 - 4$$. Multiply to get $$5 + 1 - 4$$, and simplify to find $$g(-1) = 2$$.
- **Evaluate $$g(0)$$:** Substitute $$0$$ for $$x$$ to obtain $$g(0) = 5(0)^2 - 0 - 4$$. Simplifying the exponent gives $$5(0) - 0 - 4$$, which simplifies to $$g(0) = -4$$.

Evaluating the Polynomial Function $$g(x) = 5x^2 - x - 4$$

To evaluate the polynomial function $$f(x) = 5x^2 - 8x + 4$$ for specific values, substitute the value for the independent variable $$x$$ and simplify using the order of operations: 

- **Evaluate $$f(4)$$:** Substitute $$4$$ for $$x$$ to obtain $$f(4) = 5(4)^2 - 8(4) + 4$$. Simplifying the exponent gives $$5(16) - 32 + 4$$. Multiply to get $$80 - 32 + 4$$, and simplify to find $$f(4) = 52$$.
- **Evaluate $$f(-2)$$:** Substitute $$-2$$ for $$x$$ to obtain $$f(-2) = 5(-2)^2 - 8(-2) + 4$$. Simplifying the exponent gives $$5(4) - (-16) + 4$$. Multiply to get $$20 + 16 + 4$$, and simplify to find $$f(-2) = 40$$.
- **Evaluate $$f(0)$$:** Substitute $$0$$ for $$x$$ to obtain $$f(0) = 5(0)^2 - 8(0) + 4$$. Simplifying the exponent gives $$5(0) - 0 + 4$$, which simplifies to $$f(0) = 4$$.

Evaluating the Polynomial Function $$f(x) = 5x^2 - 8x + 4$$

To evaluate the polynomial expression $$2x^2 - 5x + 3$$ when $$x = -2$$, substitute $$-2$$ for every instance of the variable $$x$$ and simplify using the order of operations. First, perform the substitution: $$2(-2)^2 - 5(-2) + 3$$. Next, evaluate the exponent: $$(-2)^2 = 4$$, which updates the expression to $$2(4) - 5(-2) + 3$$. Then, perform the multiplications from left to right: $$2(4) = 8$$ and $$-5(-2) = 10$$ (since subtracting a negative product is equivalent to adding), giving $$8 + 10 + 3$$. Finally, add the terms to find the result, which is $$21$$.

Evaluating $$2x^2 - 5x + 3$$ for $$x = -2$$

To evaluate a mathematical function when the input is a variable or an algebraic expression, follow the same procedure used for evaluating a function with a constant numerical input: substitute the given variable expression for every instance of the independent variable, such as $$x$$, in the function's defining equation. After substituting the variables, simplify the resulting algebraic expression. Unlike evaluating a function for a constant input, the final result is a new algebraic expression that represents the value of the function in terms of the new variable, rather than a single numerical value.

Evaluating a Mathematical Function for a Variable Input

A common misconception when working with function notation is assuming that a function distributes over addition, similar to the distributive property of multiplication. However, for a general mathematical function $$f$$, the evaluation of a sum is not necessarily equal to the sum of the evaluated functions. Mathematically, this means that $$f(x+a) 
eq f(x) + f(a)$$. This non-distributive property emphasizes that the function name cannot simply be distributed across terms inside the parentheses. Instead, the entire expression inside the parentheses must be substituted for the independent variable before simplifying.

Non-Distributive Property of Function Evaluation

Evaluating a mathematical function is the process of finding the numerical value of $$f(x)$$ for a specific given value of the independent variable $$x$$. To perform this process, substitute the provided value for $$x$$ into the function's equation, and then simplify the resulting numerical expression by following the order of operations.

Claude

In mathematics, equations that define functions are often written using function notation. Functions are typically assigned single-letter names, most commonly $$f$$, $$g$$, $$h$$, $$F$$, $$G$$, or $$H$$. For a function named $$f$$, the notation specifies the relationship between an input value $$x$$ and its corresponding output value $$y$$ as $$y = f(x)$$. In this notation, $$x$$ represents the domain value, and $$f(x)$$ represents the corresponding range value. The expression $$f(x)$$ is read as '$$f$$ of $$x$$' or 'the value of $$f$$ at $$x$$'. The parentheses in this context indicate the input parameter mapping to the function and do not indicate multiplication.

Function Notation

For a mathematical function expressed in the notation $$y = f(x)$$, the output variable $$y$$ (which is equivalently written as $$f(x)$$) is known as the dependent variable. It is referred to as 'dependent' because its specific value relies entirely on the value chosen for the independent variable $$x$$.

Dependent Variable of a Mathematical Function

https://openstax.org/details/books/intermediate-algebra-2e

OpenStax Intermediate Algebra

For a mathematical function expressed in the notation $$y = f(x)$$, the input variable $$x$$ is known as the independent variable. It is referred to as 'independent' because it can be assigned any valid value from the domain of the function.

Independent Variable of a Mathematical Function

Evaluating a Mathematical Function

The graph of a mathematical function is the visual representation of all its ordered pairs $$(x, y)$$ on a coordinate system. Because a function can be written using function notation where $$y = f(x)$$, these ordered pairs can equivalently be expressed as $$(x, f(x))$$. In this notation, $$f$$ represents the name of the function, $$x$$ is the $$x$$-coordinate of the ordered pair (the input), and $$f(x)$$ is the corresponding $$y$$-coordinate (the output). Thus, graphing a relation like $$f(x) = 2x - 3$$ produces the exact same line as graphing $$y = 2x - 3$$; only the notation has changed.

Graph of a Mathematical Function

For polynomial functions $$f(x)$$ and $$g(x)$$, their multiplication is defined by the formula $$(f \cdot g)(x) = f(x) \cdot g(x)$$. This means that to find the product of two functions, you multiply their respective polynomial expressions together.

Multiplication of Polynomial Functions

For polynomial functions $$f(x)$$ and $$g(x)$$, where $$g(x) 
eq 0$$, their division is defined by the formula $$(f/g)(x) = \frac{f(x)}{g(x)}$$. This means that to find the quotient function, you set up a rational expression with the polynomial for $$f(x)$$ in the numerator and the polynomial for $$g(x)$$ in the denominator. The resulting expression can then be simplified using polynomial division.

Division of Polynomial Functions

For the function $$N(t) = 75 + 10t$$, which models an account starting with $$75$$ unread emails and growing by $$10$$ emails per day, the number of unread emails is a function of the number of days. The number of unread emails, $$N$$, depends on the number of days, $$t$$. Therefore, the variable $$N$$ acts as the dependent variable, while the variable $$t$$ serves as the independent variable.

Example: Determining the Independent and Dependent Variables for the Function $$N(t) = 75 + 10t$$

In the function $$N(t) = 100 + 15t$$, modeling an account that initially has $$100$$ unread emails and accumulates $$15$$ more each day, the total number of unread emails ($$N$$) is determined by the passage of time in days ($$t$$). Since the number of emails relies on the elapsed days, $$N$$ represents the dependent variable and $$t$$ represents the independent variable.

Example: Determining the Independent and Dependent Variables for the Function $$N(t) = 100 + 15t$$

Consider the function $$N(t) = 110 + 25t$$, which describes how a baseline of $$110$$ unread emails increases by $$25$$ emails per day. In this relationship, the number of unread emails, $$N$$, is contingent upon the number of days, $$t$$. Consequently, the variable $$N$$ is the dependent variable, whereas the variable $$t$$ is the independent variable.

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