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Chapter 2: Problem 29

Let \(F(x)=x^{2}-2\) and \(G(x)=5-x .\) Find each of the following. $$(G / F)(-2)$$

Short Answer

Expert verified
\(\frac{7}{2}\)

Step by step solution

01

Understand the Composition

To find \((G / F)(-2)\), note that this represents the quotient of the functions G and F evaluated at -2. This can be written as \[\frac{G(-2)}{F(-2)}.\]
02

Evaluate G(-2)

Substitute \(-2\) into the function \G(x)=5-x. \[G(-2)=5-(-2)=5+2=7.\]
03

Evaluate F(-2)

Substitute \(-2\) into the function \F(x)=x^{2}-2. \[F(-2)=(-2)^{2}-2=4-2=2.\]
04

Compute the Quotient

Using the results from Steps 2 and 3, compute \((G / F)(-2).\) \[\frac{G(-2)}{F(-2)}=\frac{7}{2}.\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

quotient of functions
In mathematics, the quotient of functions is a term used when you divide one function by another. To find the quotient of two functions, you write the two functions as a fraction. For example, if you have two functions, F(x) and G(x), and you want to find the quotient of G and F, it would be written as \( \frac{G(x)}{F(x)} \). This allows you to understand and evaluate how one function behaves relative to another at a given point. In the provided exercise, we needed to find the quotient of the functions G and F at the value -2. This was written as \( \frac{G(-2)}{F(-2)} \).
evaluating functions
Evaluating functions means finding the value of a function for a specific input. To evaluate a function, you substitute the input into the function's formula. For instance, for the function \( G(x) = 5 - x \), if we want to find the value of G when x is -2, we substitute -2 in place of x in the formula: \( G(-2) = 5 - (-2) = 5 + 2 = 7 \). Similarly, for the function \( F(x) = x^2 - 2 \), to find F when x is -2, we substitute -2 into the formula: \( F(-2) = (-2)^2 - 2 = 4 - 2 = 2 \). These steps help simplify the overall process and are crucial to correctly solving problems involving functions.
substitution in functions
Substitution in functions is the process of replacing the variable in a function with a specific value. This is essential when evaluating functions as part of larger problems, such as finding the quotient of functions. The substitution process is straightforward: replace the variable with the given value and simplify. For example, with the function \( F(x) = x^2 - 2 \), substituting x with -2 involves changing all occurrences of x to -2, resulting in \( F(-2) = (-2)^2 - 2 = 2 \). Similarly, substituting -2 into \( G(x) = 5 - x \) results in \( G(-2) = 5 + 2 = 7 \). By following these steps, you can accurately find the values needed to solve more complex function problems.

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Most popular questions from this chapter

Graph the equations \\[ \begin{array}{l} y_{1}=1.4 x+2, \quad y_{2}=0.6 x+2 \\ y_{3}=1.4 x+5, \quad \text { and } \quad y_{4}=0.6 x+5 \end{array} \\] using a graphing calculator. If possible, use the SIMULTANEOUS mode so that you cannot tell which equation is being graphed first. Then decide which line corresponds to each equation.

Combine like terms. [ 1.3] Solve for \(m: x=\frac{m p}{c}\).

Use a graph to estimate the solution in each of the following. Be sure to use graph paper and a straightedge. Kay's Auto Village charges \(\$ 50\) for a road call plus \(\$ 15\) for each 15 -min unit of time. Estimate the time required for a road call that cost \(\$ 140\).

Explain why the domain of the function given by \(f(x)=\frac{x+3}{2}\) is \(\mathbb{R},\) but the domain of the function given by \(g(x)=\) \(\frac{2}{x+3}\) is not \(\mathbb{R}\)

Solve graphically and then check by solving algebraically. $$ 4 x-1=3-2 x $$
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