Let f(x)=1x−2 and g(x)=2x+5.
Then (f∘g)(6)=  ,
(f∘g)(x)=

Given the functions:

- \( f(x) = 1x - 2 \) (which simplifies to \( f(x) = x - 2 \))
- \( g(x) = 2x + 5 \)

### To find \( (f \circ g)(x) \):

This is the composition of functions, meaning we plug \( g(x) \) into \( f(x) \). Mathematically:

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

1. Substitute \( g(x) = 2x + 5 \) into \( f(x) = x - 2 \):

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

2. Simplify the expression:

\[
f(g(x)) = 2x + 3
\]

Thus, \( (f \circ g)(x) = 2x + 3 \).

---

### To find \( (f \circ g)(6) \):

1. First, compute \( g(6) \):

\[
g(6) = 2(6) + 5 = 12 + 5 = 17
\]

2. Now, substitute \( g(6) = 17 \) into \( f(x) = x - 2 \):

\[
f(17) = 17 - 2 = 15
\]

Thus, \( (f \circ g)(6) = 15 \).

### Final answers:
- \( (f \circ g)(x) = 2x + 3 \)
- \( (f \circ g)(6) = 15 \)

Would you like further details or have any questions?

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Here are five related questions to deepen your understanding:

1. What is the composition of \( g \circ f(x) \) in this case?
2. How would the results change if \( f(x) = 2x - 2 \) instead of \( x - 2 \)?
3. How do you interpret the composition of functions graphically?
4. What happens to \( f \circ g(x) \) when \( x = 0 \)?
5. How do you find the domain of \( (f \circ g)(x) \)?

**Tip**: When composing functions, always ensure you substitute carefully, starting from the innermost function!

Let f(x)=1x−2 and g(x)=2x+5. Then (f∘g)(6)=  , (f∘g)(x)=

Explore how to compute the function composition (f∘g)(x) where f(x) = x - 2 and g(x) = 2x + 5. This problem provides a step-by-step guide to finding (f∘g)(x) and (f∘g)(6), covering key algebraic concepts such as function composition and simplification. Suitable for students in Grades 9-11.

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