Absolutely, let's solve the problem step-by-step!
### Given functions:
[tex]\[
f(x) = 5x - 1
\][/tex]
[tex]\[
g(x) = 5x^2 - 1
\][/tex]
### Part (a): [tex]\( (f \circ g)(x) \)[/tex]
Denote [tex]\( (f \circ g)(x) \)[/tex] as [tex]\(f(g(x))\)[/tex]. This means we need to substitute [tex]\(g(x)\)[/tex] into [tex]\(f(x)\)[/tex]:
1. Find [tex]\( g(x) \)[/tex]:
[tex]\[
g(x) = 5x^2 - 1
\][/tex]
2. Substitute [tex]\(g(x)\)[/tex] into [tex]\(f(x)\)[/tex]:
[tex]\[
f(g(x)) = f(5x^2 - 1)
\][/tex]
3. Substitute [tex]\(5x^2 - 1\)[/tex] into the function [tex]\(f(x)\)[/tex]:
[tex]\[
f(5x^2 - 1) = 5(5x^2 - 1) - 1
\][/tex]
4. Simplify:
[tex]\[
f(5x^2 - 1) = 25x^2 - 5 - 1 = 25x^2 - 6
\][/tex]
Thus,
[tex]\[
(f \circ g)(x) = 25x^2 - 6
\][/tex]
### Part (b): [tex]\( (g \circ f)(x) \)[/tex]
Denote [tex]\((g \circ f)(x)\)[/tex] as [tex]\(g(f(x))\)[/tex]. This means we need to substitute [tex]\(f(x)\)[/tex] into [tex]\(g(x)\)[/tex]:
1. Find [tex]\( f(x) \)[/tex]:
[tex]\[
f(x) = 5x - 1
\][/tex]
2. Substitute [tex]\(f(x)\)[/tex] into [tex]\(g(x)\)[/tex]:
[tex]\[
g(f(x)) = g(5x - 1)
\][/tex]
3. Substitute [tex]\(5x - 1\)[/tex] into the function [tex]\(g(x)\)[/tex]:
[tex]\[
g(5x - 1) = 5(5x - 1)^2 - 1
\][/tex]
4. Simplify:
[tex]\[
(5x - 1)^2 = 25x^2 - 10x + 1
\][/tex]
[tex]\[
g(5x - 1) = 5(25x^2 - 10x + 1) - 1 = 125x^2 - 50x + 5 - 1 = 125x^2 - 50x + 4
\][/tex]
Thus,
[tex]\[
(g \circ f)(x) = 125x^2 - 50x + 4
\][/tex]
### Part (c): [tex]\( (f \circ g)(1) \)[/tex]
We found in part (a) that [tex]\( (f \circ g)(x) = 25x^2 - 6 \)[/tex]. Now we evaluate this at [tex]\( x = 1 \)[/tex]:
[tex]\[
(f \circ g)(1) = 25(1)^2 - 6 = 25 - 6 = 19
\][/tex]
Thus,
[tex]\[
(f \circ g)(1) = 19
\][/tex]
### Part (d): [tex]\( (g \circ f)(1) \)[/tex]
We found in part (b) that [tex]\( (g \circ f)(x) = 125x^2 - 50x + 4 \)[/tex]. Now we evaluate this at [tex]\( x = 1 \)[/tex]:
[tex]\[
(g \circ f)(1) = 125(1)^2 - 50(1) + 4 = 125 - 50 + 4 = 79
\][/tex]
Thus,
[tex]\[
(g \circ f)(1) = 79
\][/tex]
### Summary of Results
a. [tex]\( (f \circ g)(x) = 25x^2 - 6 \)[/tex]
b. [tex]\( (g \circ f)(x) = 125x^2 - 50x + 4 \)[/tex]
c. [tex]\( (f \circ g)(1) = 19 \)[/tex]
d. [tex]\( (g \circ f)(1) = 79 \)[/tex]
