Potter Guzman
01/13/2023 · Junior High School
11. The functions \( f, g \) and \( h \) on the set of real numbers are defined by \( f(x)=x^{2}+1 \), \( g(x)=2 x-3 \) and \( h(x)=4 x+5 \) respec- tively. Determine the formulae for the composite fuctions: \( \begin{array}{ll}\text { (a) } f[g(x)] & \text { (b) } g[f(x)] \\ \text { (c) } f h(x)] & \text { (d) } h[f(x)] \\ \text { (e) } g[h(x)] & \text { (f) } h[g(x)] \\ \text { (g) } f[g\{h(x)\}] & \text { (h) } h[g\{f(x)\}]\end{array} \)
UpStudy ThothAI Solution
Tutor-Verified Answer
Step-by-step Solution
To find the composite functions, we need to substitute the inner function into the outer function.
Given functions:
\( f(x) = x^{2} + 1 \)
\( g(x) = 2x - 3 \)
\( h(x) = 4x + 5 \)
Let's find the formulae for the composite functions:
(a) \( f[g(x)] \):
Substitute \( g(x) = 2x - 3 \) into \( f(x) = x^{2} + 1 \):
\( f[g(x)] = (2x - 3)^{2} + 1 \)
(b) \( g[f(x)] \):
Substitute \( f(x) = x^{2} + 1 \) into \( g(x) = 2x - 3 \):
\( g[f(x)] = 2(x^{2} + 1) - 3 \)
(c) \( f[h(x)] \):
Substitute \( h(x) = 4x + 5 \) into \( f(x) = x^{2} + 1 \):
\( f[h(x)] = (4x + 5)^{2} + 1 \)
(d) \( h[f(x)] \):
Substitute \( f(x) = x^{2} + 1 \) into \( h(x) = 4x + 5 \):
\( h[f(x)] = 4(x^{2} + 1) + 5 \)
(e) \( g[h(x)] \):
Substitute \( h(x) = 4x + 5 \) into \( g(x) = 2x - 3 \):
\( g[h(x)] = 2(4x + 5) - 3 \)
(f) \( h[g(x)] \):
Substitute \( g(x) = 2x - 3 \) into \( h(x) = 4x + 5 \):
\( h[g(x)] = 4(2x - 3) + 5 \)
(g) \( f[g[h(x)]] \):
Substitute \( h(x) = 4x + 5 \) into \( g(x) = 2x - 3 \), and then substitute the result into \( f(x) = x^{2} + 1 \):
\( f[g[h(x)]] = (2(4x + 5) - 3)^{2} + 1 \)
(h) \( h[g[f(x)]] \):
Substitute \( f(x) = x^{2} + 1 \) into \( g(x) = 2x - 3 \), and then substitute the result into \( h(x) = 4x + 5 \):
\( h[g[f(x)]] = 4(2(x^{2} + 1) - 3) + 5 \)
These are the formulae for the composite functions.
Quick Answer
(a) \( f[g(x)] = (2x - 3)^{2} + 1 \)
(b) \( g[f(x)] = 2(x^{2} + 1) - 3 \)
(c) \( f[h(x)] = (4x + 5)^{2} + 1 \)
(d) \( h[f(x)] = 4(x^{2} + 1) + 5 \)
(e) \( g[h(x)] = 2(4x + 5) - 3 \)
(f) \( h[g(x)] = 4(2x - 3) + 5 \)
(g) \( f[g[h(x)]] = (2(4x + 5) - 3)^{2} + 1 \)
(h) \( h[g[f(x)]] = 4(2(x^{2} + 1) - 3) + 5 \)
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