Let's solve the problem step-by-step.
### Part (a) Calculate [tex]\( f \circ g(x) \)[/tex]
The notation [tex]\( f \circ g(x) \)[/tex] means we first apply [tex]\( g(x) \)[/tex], and then apply [tex]\( f \)[/tex] to the result of [tex]\( g(x) \)[/tex]. Mathematically, this is written as [tex]\( f(g(x)) \)[/tex].
1. First, we determine [tex]\( g(x) \)[/tex]:
[tex]\[
g(x) = x + 3
\][/tex]
2. Next, we substitute [tex]\( g(x) \)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[
f(g(x)) = f(x + 3)
\][/tex]
Now, we need to apply the function [tex]\( f \)[/tex] to [tex]\( x + 3 \)[/tex]:
3. Recall that [tex]\( f(x) = x^2 + 5x \)[/tex]. Therefore, [tex]\( f(x + 3) \)[/tex] means replacing [tex]\( x \)[/tex] with [tex]\( x + 3 \)[/tex] in the function [tex]\( f \)[/tex]:
[tex]\[
f(x + 3) = (x + 3)^2 + 5(x + 3)
\][/tex]
Next, we need to simplify this expression:
4. Compute [tex]\( (x + 3)^2 \)[/tex]:
[tex]\[
(x + 3)^2 = x^2 + 6x + 9
\][/tex]
5. Compute [tex]\( 5(x + 3) \)[/tex]:
[tex]\[
5(x + 3) = 5x + 15
\][/tex]
6. Combine the results from steps 4 and 5:
[tex]\[
f(x + 3) = x^2 + 6x + 9 + 5x + 15
\][/tex]
7. Simplify the expression by combining like terms:
[tex]\[
f(x + 3) = x^2 + 11x + 24
\][/tex]
Thus:
[tex]\[
f \circ g(x) = x^2 + 11x + 24
\][/tex]
### Part (b) Calculate [tex]\( g \circ f(x) \)[/tex]
The notation [tex]\( g \circ f(x) \)[/tex] means we first apply [tex]\( f(x) \)[/tex], and then apply [tex]\( g \)[/tex] to the result of [tex]\( f(x) \)[/tex]. Mathematically, this is written as [tex]\( g(f(x)) \)[/tex].
1. First, we determine [tex]\( f(x) \)[/tex]:
[tex]\[
f(x) = x^2 + 5x
\][/tex]
2. Next, we substitute [tex]\( f(x) \)[/tex] into [tex]\( g(x) \)[/tex]:
[tex]\[
g(f(x)) = g(x^2 + 5x)
\][/tex]
Now, we need to apply the function [tex]\( g \)[/tex] to [tex]\( x^2 + 5x \)[/tex]:
3. Recall that [tex]\( g(x) = x + 3 \)[/tex]. Therefore, [tex]\( g(x^2 + 5x) \)[/tex] means replacing [tex]\( x \)[/tex] with [tex]\( x^2 + 5x \)[/tex] in the function [tex]\( g \)[/tex]:
[tex]\[
g(x^2 + 5x) = (x^2 + 5x) + 3
\][/tex]
4. Simplify the expression:
[tex]\[
g(x^2 + 5x) = x^2 + 5x + 3
\][/tex]
Thus:
[tex]\[
g \circ f(x) = x^2 + 5x + 3
\][/tex]
### Final Answers:
(a) [tex]\( f \circ g(x) = x^2 + 11x + 24 \)[/tex]
(b) [tex]\( g \circ f(x) = x^2 + 5x + 3 \)[/tex]
