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Chapter 0: Problem 57

a. If \(f(x)=x-1\) and \(h(x)=2 x+3\), find a function \(g\) such that \(h=g \circ f\). b. If \(g(x)=3 x+4\) and \(h(x)=4 x-8\), find a function \(f\) such that \(h=g \circ f\).

Short Answer

Expert verified
a. \(g(x) = 2x + 1\) b. \(f(x) = \frac{4x-40}{3}\)

Step by step solution

01

Identify the given functions

We are given the functions, \(f(x) = x-1\) and \(h(x) = 2x + 3\)
02

Rewrite h(x) using f(x)

We have to rewrite the function \(h(x)\) using \(f(x)\). So, let's replace the \(x\) in \(h(x)\) with \(f(x)\). \(h(f(x)) = 2(f(x)) + 3\)
03

Substitute f(x) into the expression

Now, we substitute \(f(x)=x-1\) into the equation: \(h(f(x)) = 2(x-1) + 3\)
04

Simplify the expression

Now, we simplify the expression: \(h(f(x)) = 2x - 2 + 3 = 2x + 1\)
05

Identify the function g(x)

Therefore, we have found the function \(g(x)\) such that \(h(x) = g(f(x))\). Thus, \(g(x) = 2x + 1\) b.
06

Identify the given functions

We are given the functions, \(g(x) = 3x+4\) and \(h(x) = 4x-8\)
07

Rewrite h(x) using g(x)

We want to express \(h(x)\) in terms of \(g(x)\). To do this, let's find the inverse of function g(x): \(g^{-1}(x)\)
08

Find the inverse of g(x)

To find the inverse, first replace \(g(x)\) with \(y\). \(y = 3x + 4\) Next, swap \(x\) and \(y\). \(x = 3y + 4\) Now, solve for \(y\): \(y = \frac{x-4}{3}\) So, \(g^{-1}(x) = \frac{x-4}{3}\)
09

Rewrite h(x) using g^{-1}(x)

Now, let's replace \(x\) in \(h(x)\) with \(g^{-1}(x)\). \(h(g^{-1}(x)) = 4(g^{-1}(x)) - 8\)
10

Substitute g^{-1}(x) into the expression

Now, substitute \(g^{-1}(x)=\frac{x-4}{3}\) into the equation: \(h(g^{-1}(x)) = 4\left(\frac{x-4}{3}\right) - 8\)
11

Simplify the expression

Now, we simplify the expression: \(h(g^{-1}(x)) = \frac{4(x-4)}{3} - 8 = \frac{4x-16}{3} - 8 = \frac{4x-16-24}{3} = \frac{4x-40}{3}\)
12

Identify the function f(x)

Therefore, we have found the function \(f(x)\) such that \(h(x) = g(f(x))\). Thus, \(f(x) = \frac{4x-40}{3}\)

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Most popular questions from this chapter

Find the exact value of the given expression. $$ \cos ^{-1} \frac{1}{2} $$

Classify each function as a polynomial function (state its degree), a power function, a rational function, an algebraic function, a trigonometric function, or other. a. \(f(x)=2 x^{3}-3 x^{2}+x-4\) b. \(f(x)=\sqrt[3]{x^{2}}\) c. \(g(x)=\frac{x}{x^{2}-4}\) d. \(f(t)=3 t^{-2}-2 t^{-1}+4\) e. \(h(x)=\frac{\sqrt{x}+1}{\sqrt{x}-1}\) f. \(f(x)=\sin x+\cos x\)

a. Show that if a function \(f\) is defined at \(-x\) whenever it is defined at \(x\), then the function \(g\) defined by \(g(x)=f(x)+f(-x)\) is an even function and the function \(h\) defined by \(h(x)=f(x)-f(-x)\) is an odd function. b. Use the result of part (a) to show that any function \(f\) defined on an interval \((-a, a)\) can be written as a sum of an even function and an odd function. c. Rewrite the function $$ f(x)=\frac{x+1}{x-1} \quad-1

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