*f*(*x*)=

−*x*−5

g, of, x, equals, x, squared, minus, 6, x, plus, 13

*g*(*x*)=

*x*2−6*x*+13

Find: , left parenthesis, f, circle, g, right parenthesis, left parenthesis, x, right parenthesis

Find: (*f*∘*g*)(*x*)

To find \((f \circ g)(x)\), we need to evaluate the composition of the functions \(f(x)\) and \(g(x)\). The composition \((f \circ g)(x)\) means that we substitute \(g(x)\) into the function \(f(x)\).

Given:
\[
f(x) = -x - 5
\]
\[
g(x) = x^2 - 6x + 13
\]

To find \((f \circ g)(x)\), substitute \(g(x)\) into \(f(x)\):

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

Substitute \(g(x)\) into \(f(x)\):

\[
f(g(x)) = -[g(x)] - 5
\]

Substitute \(g(x) = x^2 - 6x + 13\):

\[
f(g(x)) = -(x^2 - 6x + 13) - 5
\]

Distribute the negative sign:

\[
f(g(x)) = -x^2 + 6x - 13 - 5
\]

Simplify the expression:

\[
f(g(x)) = -x^2 + 6x - 18
\]

Thus, the composition \((f \circ g)(x)\) is:

\[
(f \circ g)(x) = -x^2 + 6x - 18
\]

Would you like more details or have any questions?

Here are 5 related questions:
1. How do you find the composition of two functions in general?
2. What is the domain of the composite function \((f \circ g)(x)\)?
3. How does the composition of functions relate to function transformations?
4. Can the order of composition affect the result, i.e., is \((f \circ g)(x)\) the same as \((g \circ f)(x)\)?
5. What are some practical applications of function composition in real-world problems?

**Tip:** When composing functions, always carefully substitute the inner function into the outer function and simplify step by step.

Find (f ∘ g)(x) given f(x) = -x - 5 and g(x) = x^2 - 6x + 13.

Learn how to find the composition of functions (f ∘ g)(x) with detailed steps and explanations. This example involves substituting a quadratic function into a linear function, simplifying, and understanding the algebraic manipulations required. Suitable for high school students in Grades 9-12.

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