Find f circle g (x) and g circle f for the pair of functions f(x)=(x+7)/2 and g(x)=(3-x)^(1/2)

We are asked to find \( f \circ g (x) \) and \( g \circ f (x) \) for the given functions:

1. \( f(x) = \frac{x + 7}{2} \)
2. \( g(x) = \sqrt{3 - x} \)

### 1. Finding \( f \circ g (x) \)
The notation \( f \circ g(x) \) means \( f(g(x)) \), which indicates we need to substitute \( g(x) \) into \( f(x) \).

\[
f(g(x)) = f\left( \sqrt{3 - x} \right)
\]
Substitute \( \sqrt{3 - x} \) into \( f(x) = \frac{x + 7}{2} \):

\[
f\left( \sqrt{3 - x} \right) = \frac{\sqrt{3 - x} + 7}{2}
\]

Thus:
\[
f \circ g (x) = \frac{\sqrt{3 - x} + 7}{2}
\]

### 2. Finding \( g \circ f (x) \)
The notation \( g \circ f(x) \) means \( g(f(x)) \), so we need to substitute \( f(x) \) into \( g(x) \).

\[
g(f(x)) = g\left( \frac{x + 7}{2} \right)
\]
Substitute \( \frac{x + 7}{2} \) into \( g(x) = \sqrt{3 - x} \):

\[
g\left( \frac{x + 7}{2} \right) = \sqrt{3 - \frac{x + 7}{2}}
\]

Simplify inside the square root:

\[
g\left( \frac{x + 7}{2} \right) = \sqrt{\frac{6 - (x + 7)}{2}} = \sqrt{\frac{-x - 1}{2}}
\]

Since this expression involves a negative value under the square root for certain values of \( x \), this composition is not defined for all \( x \). It is only defined when \( -x - 1 \geq 0 \), meaning \( x \leq -1 \).

Thus:
\[
g \circ f (x) = \sqrt{\frac{-x - 1}{2}}, \quad \text{for } x \leq -1
\]

### Summary:
- \( f \circ g (x) = \frac{\sqrt{3 - x} + 7}{2} \)
- \( g \circ f (x) = \sqrt{\frac{-x - 1}{2}}, \text{ for } x \leq -1 \)

Do you have any questions, or would you like more details?

Here are 5 related questions for further exploration:
1. What is the domain of \( f \circ g (x) \)?
2. What is the domain of \( g \circ f (x) \)?
3. Can the function \( f \circ g (x) \) have any discontinuities?
4. How would you graph \( f \circ g (x) \)?
5. Can you find the inverse of \( f(x) \) or \( g(x) \)?

**Tip:** When combining functions, always check the domain of the resulting composition to avoid undefined expressions.

Learn how to find the compositions f(g(x)) and g(f(x)) for the functions f(x) = (x+7)/2 and g(x) = sqrt(3-x). This problem explores key algebraic concepts, including function composition and domain restrictions. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation