https://file.aiquickdraw.com/mathbot/file/1727041000128ba1l0sxx.jpg

We are given that the function \( f(x) = x^2 + 4 \), and we need to find a function \( g(x) \) such that \( (f \circ g)(x) = x + 8 \).

To solve this, recall that \( (f \circ g)(x) = f(g(x)) \), which means we must have:

\[
f(g(x)) = x + 8
\]

Since \( f(x) = x^2 + 4 \), we substitute \( g(x) \) into this equation:

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

Now, we solve the equation:

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

Subtract 4 from both sides:

\[
(g(x))^2 = x + 4
\]

Now take the square root of both sides to solve for \( g(x) \):

\[
g(x) = \pm \sqrt{x + 4}
\]

Thus, the function \( g(x) \) could be either \( g(x) = \sqrt{x + 4} \) or \( g(x) = -\sqrt{x + 4} \).

Would you like more details or have any questions?

Here are some related questions to explore:

1. How does the composition of functions work in general?
2. What happens if we change \( f(x) \) to another form?
3. How can we verify which solution, \( +\sqrt{x+4} \) or \( -\sqrt{x+4} \), fits better in other contexts?
4. How does the domain of \( f(x) \) and \( g(x) \) affect the overall solution?
5. What if \( (f \circ g)(x) \) were set equal to something different, like \( x + 10 \)?

**Tip**: Always check the domains of functions after finding compositions, especially when dealing with square roots.

The function f is given by f(x) = x^2 + 4. Find a function g with (f ∘ g)(x) = x + 8.

Learn how to find a function g(x) such that (f ∘ g)(x) = x + 8 for the function f(x) = x^2 + 4. This problem involves function composition and solving quadratic equations. 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