Problem: Solve the composite function problems from the worksheet.
####
Part 1: Evaluate each composite value
1.
If \( f(x) = 3x - 5 \) and \( g(x) = x^2 \), find \( (f \circ g)(3) \)
-
Step 1: Compute \( g(3) \):
\[
g(3) = 3^2 = 9
\]
-
Step 2: Substitute \( g(3) \) into \( f(x) \):
\[
f(g(3)) = f(9) = 3(9) - 5 = 27 - 5 = 22
\]
-
Answer:
\[
\boxed{22}
\]
2.
If \( f(x) = -9x - 9 \) and \( g(x) = \sqrt{x - 9} \), find \( (f \circ g)(10) \)
-
Step 1: Compute \( g(10) \):
\[
g(10) = \sqrt{10 - 9} = \sqrt{1} = 1
\]
-
Step 2: Substitute \( g(10) \) into \( f(x) \):
\[
f(g(10)) = f(1) = -9(1) - 9 = -9 - 9 = -18
\]
-
Answer:
\[
\boxed{-18}
\]
3.
If \( f(x) = -4x + 2 \) and \( g(x) = \sqrt{x - 8} \), find \( (f \circ g)(12) \)
-
Step 1: Compute \( g(12) \):
\[
g(12) = \sqrt{12 - 8} = \sqrt{4} = 2
\]
-
Step 2: Substitute \( g(12) \) into \( f(x) \):
\[
f(g(12)) = f(2) = -4(2) + 2 = -8 + 2 = -6
\]
-
Answer:
\[
\boxed{-6}
\]
4.
If \( f(x) = -3x + 4 \) and \( g(x) = x^2 \), find \( (g \circ f)(-2) \)
-
Step 1: Compute \( f(-2) \):
\[
f(-2) = -3(-2) + 4 = 6 + 4 = 10
\]
-
Step 2: Substitute \( f(-2) \) into \( g(x) \):
\[
g(f(-2)) = g(10) = 10^2 = 100
\]
-
Answer:
\[
\boxed{100}
\]
5.
If \( f(x) = -2x + 1 \) and \( g(x) = \sqrt{x^2 - 5} \), find \( (g \circ f)(2) \)
-
Step 1: Compute \( f(2) \):
\[
f(2) = -2(2) + 1 = -4 + 1 = -3
\]
-
Step 2: Substitute \( f(2) \) into \( g(x) \):
\[
g(f(2)) = g(-3) = \sqrt{(-3)^2 - 5} = \sqrt{9 - 5} = \sqrt{4} = 2
\]
-
Answer:
\[
\boxed{2}
\]
####
Part 2: Find each composite
6.
Given \( f(x) = -9x + 3 \) and \( g(x) = x^4 \), find \( (f \circ g)(x) \)
-
Step 1: Substitute \( g(x) \) into \( f(x) \):
\[
f(g(x)) = f(x^4) = -9(x^4) + 3 = -9x^4 + 3
\]
-
Answer:
\[
\boxed{-9x^4 + 3}
\]
7.
Given \( f(x) = 2x - 5 \) and \( g(x) = x + 2 \), find \( (f \circ g)(x) \)
-
Step 1: Substitute \( g(x) \) into \( f(x) \):
\[
f(g(x)) = f(x + 2) = 2(x + 2) - 5 = 2x + 4 - 5 = 2x - 1
\]
-
Answer:
\[
\boxed{2x - 1}
\]
Final Answers:
1. \(\boxed{22}\)
2. \(\boxed{-18}\)
3. \(\boxed{-6}\)
4. \(\boxed{100}\)
5. \(\boxed{2}\)
6. \(\boxed{-9x^4 + 3}\)
7. \(\boxed{2x - 1}\)
Parent Tip: Review the logic above to help your child master the concept of composition functions worksheet.
