IDNLearn.com provides a collaborative environment for finding accurate answers. Our experts are available to provide accurate, comprehensive answers to help you make informed decisions about any topic or issue you encounter.
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]
Thank you for joining our conversation. Don't hesitate to return anytime to find answers to your questions. Let's continue sharing knowledge and experiences! IDNLearn.com provides the answers you need. Thank you for visiting, and see you next time for more valuable insights.
