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Chapter 9: Problem 30

For each pair of functions \(f\) and \(g\), find \((a) f+g,\) (b) \(f-g,\) (c) \(f g\), and \((d) \frac{f}{g}\). Give the domain for each. See Example 2. $$ f(x)=-2 x+9, \quad g(x)=-5 x+2 $$

Short Answer

Expert verified
a:\(-7x + 11\), b:\(3x + 7\), c:\(10x{^2} - 49x + 18\), d:\( \frac{-2x + 9}{-5x + 2} \) excluding \( x \ = \ \ \frac{2}{5} \

Step by step solution

01

Add the Functions

To find \( f + g \), sum the expressions for \( f(x) \) and \( g(x) \). \[ f(x) = -2x + 9 \] \[ g(x) = -5x + 2 \] Thus, \[ f + g = (-2x + 9) + (-5x + 2) \] Simplify this to get: \[ f + g = -7x + 11 \] The domain of \( f + g \) is all real numbers, since both \( f(x) \) and \( g(x) \) are polynomials.
02

Subtract the Functions

To find \( f - g \), subtract the expression of \( g(x) \) from \( f(x) \). \[ f(x) = -2x + 9 \] \[ g(x) = -5x + 2 \] Thus, \[ f - g = (-2x + 9) - (-5x + 2) \] Simplify this to get: \[ f - g = -2x + 9 + 5x - 2 = 3x + 7 \] The domain of \( f - g \) is all real numbers, as both \( f(x) \) and \( g(x) \) are polynomials.
03

Multiply the Functions

To find \( f \cdot g \), multiply the expressions for \( f(x) \) and \( g(x) \). \[ f(x) = -2x + 9 \] \[ g(x) = -5x + 2 \] Thus, \[ f \cdot g = (-2x + 9)(-5x + 2) \] Expand this product: \[ f \cdot g = 10x^2 - 4x - 45x + 18 \] Simplify to get: \[ f \cdot g = 10x^2 - 49x + 18 \] The domain of \( f \cdot g \) is all real numbers, as polynomials have a domain of all real numbers.
04

Divide the Functions

To find \( \frac{f}{g} \), divide \( f(x) \) by \( g(x) \). \[ f(x) = -2x + 9 \] \[ g(x) = -5x + 2 \] Thus, \[\frac{f}{g} = \frac{-2x + 9}{-5x + 2} \] The domain of \( \frac{f}{g} \) includes all real numbers except where \( g(x) = 0. \) Therefore, solve for when \[ -5x + 2 = 0 \] \[ x = \frac{2}{5} \] Thus, the domain of \( \frac{f}{g} \) is all real numbers except \ x = \frac{2}{5} \.

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Key Concepts

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

Function Addition
Function addition is straightforward. To add the functions \(f(x)\) and \(g(x)\), you sum their respective expressions. For example, given \(f(x) = -2x + 9\) and \(g(x) = -5x + 2\), you simply combine like terms:

\[ f + g = (-2x + 9) + (-5x + 2) = -7x + 11 \]
This shows the new function formed by adding \(f(x)\) and \(g(x)\). Because both functions are polynomials, the domain is all real numbers. This means any real number can be inputted into the function without restriction.
Function Subtraction
Function subtraction means subtracting the second function \(g(x)\) from the first function \(f(x)\). Using the functions \(f(x) = -2x + 9\) and \(g(x) = -5x + 2\), subtract their expressions and simplify:

\[ f - g = (-2x + 9) - (-5x + 2) = -2x + 9 + 5x - 2 = 3x + 7 \]
As with addition, the result is a new function with a domain of all real numbers, because polynomials have no domain restrictions.
Function Multiplication
To multiply two functions, you multiply their expressions together. For \(f(x) = -2x + 9\) and \(g(x) = -5x + 2\), do the following:

\[ f \times g = (-2x + 9)(-5x + 2) = 10x^2 - 4x - 45x + 18 = 10x^2 - 49x + 18 \]
By expanding and combining like terms, you form a new polynomial. As usual, this polynomial's domain is all real numbers.
Function Division
Function division involves dividing one function by the other. Given \(f(x) = -2x + 9\) and \(g(x) = -5x + 2\), set up the division and solve:

\[ \frac{f}{g} = \frac{-2x + 9}{-5x + 2} \]
However, ensure there are no division by zero errors. This happens when the denominator \(g(x)\) equals zero:

\[ -5x + 2 = 0 \rightarrow x = \frac{2}{5} \]
Thus, the domain of \( \frac{f}{g} \) is all real numbers except \( x = \frac{2}{5} \).

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Most popular questions from this chapter

The tables give some selected ordered pairs for functions \(f\) and \(g\). $$\begin{array}{r|r}{x} & f(x) \\\\\hline-1 & 1 \\\\\hline 2 & -1 \\\\\hline 5 & 9\end{array}$$ $$\begin{array}{c|c}x & g(x) \\\\\hline 2 & -1 \\\\\hline 7 & 5 \\\\\hline 1 & 9 \\\\\hline 9 & 20\end{array}$$ Find each of the following. $$ (g \circ f)(5) $$

Graph each absolute value function. \(f(x)=|-3-x|\)

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The tables give some selected ordered pairs for functions \(f\) and \(g\). $$\begin{array}{r|r}{x} & f(x) \\\\\hline-1 & 1 \\\\\hline 2 & -1 \\\\\hline 5 & 9\end{array}$$ $$\begin{array}{c|c}x & g(x) \\\\\hline 2 & -1 \\\\\hline 7 & 5 \\\\\hline 1 & 9 \\\\\hline 9 & 20\end{array}$$ Find each of the following. $$ (f \circ f)(2) $$
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