To find the quotient function and evaluate it for the functions $$f(x) = x^2 - 5x - 14$$ and $$g(x) = x + 2$$, follow two steps: Step 1 — Find $$(f/g)(x)$$. Apply the definition of division for polynomial functions: $$(f/g)(x) = \frac{f(x)}{g(x)}$$. Substitute the given polynomials to form the rational expression $$\frac{x^2 - 5x - 14}{x + 2}$$. Use polynomial division to divide the numerator by the denominator. Dividing $$x^2 - 5x - 14$$ by $$x + 2$$ yields $$x - 7$$. Therefore, $$(f/g)(x) = x - 7$$. Step 2 — Evaluate $$(f/g)(-4)$$. Substitute the specific value $$-4$$ for $$x$$ in the resulting quotient function: $$(f/g)(-4) = (-4) - 7 = -11$$.

Finding the Quotient of Polynomial Functions $$(x^2 - 5x - 14) \div (x + 2)$$ and Evaluating at $$x = -4$$

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.

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

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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

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

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

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.

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