Question
Refer to the functions \( f, g \), and \( h \) and find the given function. \[ f(x)=3 x+5 \] \[ g(x)=x^{2} \] \( (f \circ g \circ h)(x)=\square \)
Ask by Pritchard Mckenzie. in the United States
Mar 10,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\[
(f \circ g \circ h)(x) = 3 \, (h(x))^2 + 5
\]
Solution
We need to compute the composition
\[
(f \circ g \circ h)(x) = f(g(h(x))).
\]
**Step 1. Apply \( g \) to \( h(x) \):**
Since
\[
g(x) = x^2,
\]
we have
\[
g(h(x)) = (h(x))^2.
\]
**Step 2. Apply \( f \) to \( g(h(x)) \):**
Given
\[
f(x) = 3x + 5,
\]
substitute \( x = (h(x))^2 \) into \( f \):
\[
f\bigl((h(x))^2\bigr) = 3 \,(h(x))^2 + 5.
\]
Thus, the composite function is
\[
(f \circ g \circ h)(x) = 3 \,(h(x))^2 + 5.
\]
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Simplify this solution Bonus Knowledge
To find the composite function \( (f \circ g \circ h)(x) \), we need to substitute \( g(x) \) into \( f(x) \) and also consider the function \( h(x) \). Let's denote \( h(x) = x \) for simplicity. First, we calculate \( g(h(x)) \): \[ g(h(x)) = g(x) = x^2. \] Now, substitute \( g(h(x)) \) into \( f(x) \): \[ f(g(h(x))) = f(g(x)) = f(x^2). \] Using the function \( f(x) = 3x + 5 \), we have: \[ f(x^2) = 3(x^2) + 5 = 3x^2 + 5. \] Thus, the composite function is: \[ (f \circ g \circ h)(x) = 3x^2 + 5. \]
