composition on function f(x)= x-4
g(x)= x^2-8x+16

To find the composition of two functions, we need to determine \( (f \circ g)(x) \) and \( (g \circ f)(x) \), which represent \( f(g(x)) \) and \( g(f(x)) \), respectively.

### 1. \( (f \circ g)(x) = f(g(x)) \)
Given:
- \( f(x) = x - 4 \)
- \( g(x) = x^2 - 8x + 16 \)

To compute \( f(g(x)) \), substitute \( g(x) \) into \( f(x) \):

\[
f(g(x)) = f(x^2 - 8x + 16) = (x^2 - 8x + 16) - 4
\]

Simplifying:

\[
f(g(x)) = x^2 - 8x + 12
\]

### 2. \( (g \circ f)(x) = g(f(x)) \)
Now, let's compute \( g(f(x)) \) by substituting \( f(x) \) into \( g(x) \):

\[
g(f(x)) = g(x - 4) = (x - 4)^2 - 8(x - 4) + 16
\]

First, expand \( (x - 4)^2 \):

\[
(x - 4)^2 = x^2 - 8x + 16
\]

Now expand \( -8(x - 4) \):

\[
-8(x - 4) = -8x + 32
\]

Substitute these into the equation:

\[
g(f(x)) = (x^2 - 8x + 16) + (-8x + 32) + 16
\]

Simplify:

\[
g(f(x)) = x^2 - 16x + 64
\]

### Final Results:
- \( f(g(x)) = x^2 - 8x + 12 \)
- \( g(f(x)) = x^2 - 16x + 64 \)

Would you like further details or have any questions?

Here are 5 related questions you might explore:
1. What happens to the compositions if we change the functions slightly?
2. How do you graph \( f(g(x)) \) and \( g(f(x)) \)?
3. Can we find the domain of \( f(g(x)) \) and \( g(f(x)) \)?
4. What if we compose more than two functions, like \( f(g(h(x))) \)?
5. How does composition differ from addition and multiplication of functions?

**Tip:** When composing functions, always remember to substitute the entire second function into the first one, being careful with parentheses and order of operations!

composition on function f(x)= x-4, g(x)= x^2-8x+16

Explore the function composition of f(x) = x-4 and g(x) = x^2-8x+16. Learn how to calculate f(g(x)) and g(f(x)) with detailed step-by-step explanations. This problem is ideal for students in Grades 9-11, focusing on algebraic function operations.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation