f(x)=7-4x and g(x)=2+x^2     (f ∘ g)(x)=

To find \((f \circ g)(x)\), also known as the composition of functions \(f\) and \(g\), you substitute \(g(x)\) into \(f(x)\). In other words:

\[
(f \circ g)(x) = f(g(x))
\]

Given that:
- \(f(x) = 7 - 4x\)
- \(g(x) = 2 + x^2\)

Now, substitute \(g(x)\) into \(f(x)\):

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

Substitute \(2 + x^2\) into \(f(x)\):

\[
f(2 + x^2) = 7 - 4(2 + x^2)
\]

Simplify the expression:

\[
f(2 + x^2) = 7 - 4(2) - 4(x^2)
\]
\[
= 7 - 8 - 4x^2
\]
\[
= -1 - 4x^2
\]

Thus, the composition of \(f\) and \(g\) is:

\[
(f \circ g)(x) = -1 - 4x^2
\]

Would you like further details on function composition or any other clarification?

Here are 5 related questions to consider:
1. How would you compute \((g \circ f)(x)\)?
2. What happens if \(f(x)\) and \(g(x)\) are inverse functions?
3. Can you find the domain of \((f \circ g)(x)\)?
4. How does function composition affect the range of the resulting function?
5. Can you graph \(f(x)\), \(g(x)\), and \((f \circ g)(x)\) to visualize the result?

**Tip:** In function composition, pay careful attention to the order of the functions. \(f(g(x))\) is not the same as \(g(f(x))\).

Learn how to find the composition of functions f(x)=7-4x and g(x)=2+x^2. This example explains key algebraic concepts and demonstrates how to substitute one function into another to solve for (f ∘ g)(x). Suitable for high school students in Grades 9-12.

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