Problem 95 - Precalculus Essentials

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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\.

Step 2: 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 \(x\): \(2x^2-6x+11=7\).

Step 3: 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 \(x=1\).