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Chapter 2: Problem 6

Find \(f(g(x))\) and \(g(f(x))\) and determine whether each pair of functions \(f\) and \(g\) are inverses of each other. $$ f(x)=3 x-7 \text { and } g(x)=\frac{x+3}{7} $$

Short Answer

Expert verified
The functions \(f\) and \(g\) are inverses of each other.

Step by step solution

01

Finding \(f(g(x))\)

We substitute the function \(g(x)\) into function \(f(x)\). So \(f(g(x)) = f(\frac{x+3}{7}) = 3*(\frac{x+3}{7}) - 7 = \frac{3x + 9}{7} - 7 = \frac{3x + 9 - 49}{7} = \frac{3x - 40}{7} = x.\
02

Finding \(g(f(x))\)

Now, substitute function \(f(x)\) into function \(g(x)\). So \(g(f(x)) = g(3x - 7) = \frac{3x-7 + 3}{7} = \frac{3x - 4}{7} = x.\
03

Checking inverse property

Both compositions equal \(x\), which means according to the property of inverse functions, \(f\) and \(g\) are inverses of each other.

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Most popular questions from this chapter

Find a. \((f \circ g)(x)\) b. \((g \circ f)(x)\) c. \((f \circ g)(2)\) d. \((g \circ f)(2)\) $$f(x)=\sqrt{x}, g(x)=x+2$$

Find all values of x satisfying the given conditions. $$f(x)=1-2 x, g(x)=3 x^{2}+x-1, \text { and }(f \circ g)(x)=-5$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. If \(f(x)=\sqrt{x}\) and \(g(x)=2 x-1,\) then \((f \circ g)(5)=g(2)\)

Complete the square and write the equation in standard form. Then give the center and radius of each circle and graph the equation. $$x^{2}+y^{2}+3 x+5 y+\frac{9}{4}=0$$

Solve and check: \(\frac{x-1}{5}-\frac{x+3}{2}=1-\frac{x}{4}\) (Section \(1.2, \text { Example } 3)\)
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