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Sagot :
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]
### 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]
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