Question

for f(x)=5x−1 and g(x)=5x²−1, find the following functions. a. (f∘g)(x); b. (g∘f)(x); c. (f∘g)(2); d. (g∘f)(2) a. (f∘g)(x)=□ (simplify your answer.) b. (g∘f)(x)=□ (simplify your answer.) c. (f∘g)(2)=□ (simplify your answer.) d. (g∘f)(2)=□ (simplify your answer.)

Explanation:

Step1: Find \((f \circ g)(x)\)

The composition \((f \circ g)(x)\) means \(f(g(x))\). Substitute \(g(x)=5x^{2}-1\) into \(f(x)\). So \(f(g(x)) = f(5x^{2}-1)\). Since \(f(x)=5x - 1\), replace \(x\) with \(5x^{2}-1\): \(f(5x^{2}-1)=5(5x^{2}-1)-1\).
Simplify: \(5\times5x^{2}-5\times1 - 1=25x^{2}-5 - 1=25x^{2}-6\).

Step2: Find \((g \circ f)(x)\)

The composition \((g \circ f)(x)\) means \(g(f(x))\). Substitute \(f(x)=5x - 1\) into \(g(x)\). So \(g(f(x))=g(5x - 1)\). Since \(g(x)=5x^{2}-1\), replace \(x\) with \(5x - 1\): \(g(5x - 1)=5(5x - 1)^{2}-1\).
Expand \((5x - 1)^{2}=(5x)^{2}-2\times5x\times1 + 1^{2}=25x^{2}-10x + 1\). Then \(5(25x^{2}-10x + 1)-1=125x^{2}-50x + 5 - 1=125x^{2}-50x + 4\).

Step3: Find \((f \circ g)(2)\)

We can use the result from part (a). Substitute \(x = 2\) into \((f \circ g)(x)=25x^{2}-6\). So \((f \circ g)(2)=25\times(2)^{2}-6\).
Calculate: \(25\times4-6 = 100 - 6=94\).

Step4: Find \((g \circ f)(2)\)

We can use the result from part (b). Substitute \(x = 2\) into \((g \circ f)(x)=125x^{2}-50x + 4\). So \((g \circ f)(2)=125\times(2)^{2}-50\times2 + 4\).
Calculate: \(125\times4-100 + 4=500-100 + 4 = 404\).

Answer:

s:
a. \((f \circ g)(x)=\boldsymbol{25x^{2}-6}\)
b. \((g \circ f)(x)=\boldsymbol{125x^{2}-50x + 4}\)
c. \((f \circ g)(2)=\boldsymbol{94}\)
d. \((g \circ f)(2)=\boldsymbol{404}\)