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Chapter 1: Problem 113

Use the functions \(f(x)=x+4\) and \(g(x)=2 x-5\) to find the specified function. $$(f \circ g)^{-1}$$

Short Answer

Expert verified
The inverse of the function obtained by the composition of \(f(x)=x+4\) and \(g(x)=2x-5\) is \((f \circ g)^{-1}(x)=(x+1)/2\).

Step by step solution

01

Compose the Functions

Apply the functions sequentially as specified by the composition \(f \circ g\). That means to apply function \(g\) first, then function \(f\). In this case, substitute \(g(x)=2x-5\) into \(f(x)=x+4\). Thus, the composition \(f(g(x))=f(2x-5)=(2x-5) + 4 = 2x-1\).
02

Compute the Inverse Function

To find the inverse of \(f(g(x))=2x-1\), we replace \(f(g(x))\) with \(y\) to get \(y=2x-1\). Interchange \(x\) and \(y\) to get \(x=2y-1\), then solve for \(y\) to obtain the inverse function: Adding 1 and dividing by 2 gives the inverse function \((f \circ g)^{-1}(x)=(x+1)/2\).
03

Check the Inverse Function

To confirm that the function obtained is truly the inverse, we can check by verifying whether \((f \circ g) \circ (f \circ g)^{-1}(x)=x\) and \((f \circ g)^{-1} \circ (f \circ g)(x)=x\). Substituting \((f \circ g)^{-1}(x)=(x+1)/2\) into the equation results in \((f \circ g) \circ (f \circ g)^{-1}(x)=2((x+1)/2) -1 = x\), and \((f \circ g)^{-1} \circ (f \circ g)(x)=((2x-1)+1)/2=x\). Thus, the function obtained is indeed the inverse.

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Most popular questions from this chapter

Graph the function and determine the interval(s) (if any) on the real axis for which \(f(x) \geq 0\) Use a graphing utility to verify your results. $$f(x)=4-x$$

Determine whether the lines \(L_{1}\) and \(L_{2}\) passing through the pairs of points are parallel, perpendicular, or neither.$$\begin{aligned}&L_{1}:(-1,-7),(4,3)\\\&L_{2}:(1,5),(-2,-7)\end{aligned}$$.

You can encode and decode messages using functions and their inverses. To code a message, first translate the letters to numbers using 1 for "A," 2 for "B," and so on. Use 0 for a space. So, "A ball" becomes 1 0 2 1 12 12. Then, use a one-to-one function to convert to coded numbers. Using \(f(x)=2 x-1,\) "A ball" becomes 1 ?1 3 1 23 23. (a) Encode "Call me later" using the function \(f(x)=5 x+4.\) (b) Find the inverse function of \(f(x)=5 x+4\) and use it to decode 119 44 9 104 4 104 49 69 29.

The table shows men's shoe sizes in the United States and the corresponding European shoe sizes. Let \(y=f(x)\) represent the function that gives the men's European shoe size in terms of \(x,\) the men's U.S. size. $$\begin{array}{|c|c|}\hline \text { Men鈥檚 U.S. } & \text { Men鈥檚 European } \\\\\text { shoe size } & \text { shoe size } \\\\\hline 8 & 41 \\\9 & 42 \\\10 & 43 \\\11 & 44 \\\12 & 45 \\\13 & 46 \\\\\hline\end{array}$$ (a) Is \(f\) one-to-one? Explain. (b) Find \(f(11)\). (c) Find \(f^{-1}(43),\) if possible. (d) Find \(f\left(f^{-1}(41)\right)\). (e) Find \(f^{-1}(f(12))\).

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