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Chapter 1: Problem 95

Find all values of \(x\) satisfying the given conditions. $$f(x)=2 x-5, g(x)=x^{2}-3 x+8, \text { and }(f \circ g)(x)=7$$

Short Answer

Expert verified
The values of \(x\) that satisfy the given condition are \(x=1\) and \(x=2\).

Step by step solution

01

Find the composition function \(f \circ g\)

To find \(f(g(x))\) (which is the same as \(f \circ g(x)\)), replace \(x\) in function \(f\) with \(g(x)\). That is, \(f(g(x))=2(g(x))-5=2(x^2-3x+8)-5 = 2x^2-6x+16-5=2x^2-6x+11\.
02

Equate the composition function to 7

(f \circ g)(x) has to be equal to 7, hence write down the equation and solve it for \(x\): \(2x^2-6x+11=7\).
03

Solve for \(x\)

Subtract 7 from both sides to set the equation to zero: \(2x^2-6x+11-7=0\), which simplifies further to \(2x^2-6x+4=0\). To make the equation simpler, divide each term by 2 to get \(x^2 - 3x + 2 = 0\). Now, factor the quadratic equation to get \((x-2)(x-1)=0\). Setting each factor equal to zero gives the solutions: if \(x-2=0\) then \(x=2\) and if \(x-1=0\) then \(x=1\).

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