Certainly! Let's solve each part step by step using the given functions [tex]\( f(x) = 5x - 3 \)[/tex] and [tex]\( g(x) = |x| \)[/tex].
### Part (a): [tex]\((f \circ g)(-3)\)[/tex]
The composition [tex]\((f \circ g)(x)\)[/tex] means we first apply [tex]\(g(x)\)[/tex], and then apply [tex]\(f\)[/tex] to the result.
1. Apply [tex]\(g(x)\)[/tex] to [tex]\(-3\)[/tex]:
The function [tex]\(g(x)\)[/tex] takes the absolute value of [tex]\(x\)[/tex].
[tex]\[
g(-3) = |-3| = 3
\][/tex]
2. Apply [tex]\(f(x)\)[/tex] to the result from [tex]\(g(-3)\)[/tex]:
Now we use the result [tex]\(3\)[/tex] from step 1 and plug it into the function [tex]\(f(x)\)[/tex].
[tex]\[
f(3) = 5 \cdot 3 - 3
\][/tex]
3. Simplify the result:
[tex]\[
f(3) = 15 - 3 = 12
\][/tex]
Therefore, [tex]\((f \circ g)(-3) = 12\)[/tex].
Answer:
[tex]\[
(f \circ g)(-3) = 12
\][/tex]
### Part (b): [tex]\((g \circ f)(4)\)[/tex]
The composition [tex]\((g \circ f)(x)\)[/tex] means we first apply [tex]\(f(x)\)[/tex], and then apply [tex]\(g\)[/tex] to the result.
1. Apply [tex]\(f(x)\)[/tex] to [tex]\(4\)[/tex]:
The function [tex]\(f(x)\)[/tex] is given by [tex]\(f(x) = 5x - 3\)[/tex].
[tex]\[
f(4) = 5 \cdot 4 - 3
\][/tex]
2. Simplify the result from [tex]\(f(4)\)[/tex]:
[tex]\[
f(4) = 20 - 3 = 17
\][/tex]
3. Apply [tex]\(g(x)\)[/tex] to the result from [tex]\(f(4)\)[/tex]:
Now we use the result [tex]\(17\)[/tex] from step 2 and plug it into the function [tex]\(g(x)\)[/tex].
[tex]\[
g(17) = |17| = 17
\][/tex]
Therefore, [tex]\((g \circ f)(4) = 17\)[/tex].
Answer:
[tex]\[
(g \circ f)(4) = 17
\][/tex]
This concludes the detailed step-by-step solution for both parts.
