Questions: For f(x)=x^2+5 and g(x)=x^2-5, find the following functions. a. (f ∘ g)(x); b. (g ∘ f)(x); c. (f ∘ g)(3); d. (g ∘ f)(3) a. (f ∘ g)(x)=

For f(x)=x^2+5 and g(x)=x^2-5, find the following functions.
a. (f ∘ g)(x);
b. (g ∘ f)(x);
c. (f ∘ g)(3);
d. (g ∘ f)(3)
a. (f ∘ g)(x)=

Transcript text: For $f(x)=x^{2}+5$ and $g(x)=x^{2}-5$, find the following functions. a. $(f \circ g)(x)$; b. $(g \circ f)(x)$; c. $(f \circ g)(3)$; d. $(g \circ f)(3)$ a. $(f \circ g)(x)=$ $\square$

Solution

Solution Steps

Step 1: Understand the Composition of Functions

The composition of functions \( (f \circ g)(x) \) means applying \( g(x) \) first and then applying \( f \) to the result. Similarly, \( (g \circ f)(x) \) means applying \( f(x) \) first and then applying \( g \) to the result.

Step 2: Calculate \( (f \circ g)(x) \)

Given:

\( f(x) = x^2 + 5 \)
\( g(x) = x^2 - 5 \)

To find \( (f \circ g)(x) \), substitute \( g(x) \) into \( f(x) \):

\[ (f \circ g)(x) = f(g(x)) = f(x^2 - 5) = (x^2 - 5)^2 + 5 \]

Now, expand \( (x^2 - 5)^2 \):

\[ (x^2 - 5)^2 = x^4 - 10x^2 + 25 \]

Substitute back into the expression for \( f(g(x)) \):

\[ f(g(x)) = x^4 - 10x^2 + 25 + 5 = x^4 - 10x^2 + 30 \]

Step 3: Calculate \( (g \circ f)(x) \)

To find \( (g \circ f)(x) \), substitute \( f(x) \) into \( g(x) \):

\[ (g \circ f)(x) = g(f(x)) = g(x^2 + 5) = (x^2 + 5)^2 - 5 \]

Now, expand \( (x^2 + 5)^2 \):

\[ (x^2 + 5)^2 = x^4 + 10x^2 + 25 \]

Substitute back into the expression for \( g(f(x)) \):

\[ g(f(x)) = x^4 + 10x^2 + 25 - 5 = x^4 + 10x^2 + 20 \]

Step 4: Calculate \( (f \circ g)(3) \)

First, find \( g(3) \):

\[ g(3) = 3^2 - 5 = 9 - 5 = 4 \]

Now, find \( f(g(3)) = f(4) \):

\[ f(4) = 4^2 + 5 = 16 + 5 = 21 \]

Final Answer

\( (f \circ g)(x) = \boxed{x^4 - 10x^2 + 30} \)
\( (g \circ f)(x) = \boxed{x^4 + 10x^2 + 20} \)
\( (f \circ g)(3) = \boxed{21} \)

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