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.

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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.

Evaluating a Mathematical Function

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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

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.

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