Suppose 
f
(
x
)
=
√
x
2
−
3
x
+
6
 and 
g
(
x
)
=
−
4
x
+
6
.

(
f
∘
g
)
(
1
)

We are asked to find \( (f \circ g)(1) \), which means evaluating \( f(g(1)) \).

Given:
- \( f(x) = \sqrt{x^2 - 3x + 6} \)
- \( g(x) = -4x + 6 \)

### Step 1: Find \( g(1) \)
To find \( g(1) \), substitute \( x = 1 \) into \( g(x) \):
\[
g(1) = -4(1) + 6 = -4 + 6 = 2
\]

### Step 2: Find \( f(g(1)) = f(2) \)
Now substitute \( g(1) = 2 \) into \( f(x) \):
\[
f(2) = \sqrt{2^2 - 3(2) + 6}
\]
Simplify the expression inside the square root:
\[
f(2) = \sqrt{4 - 6 + 6} = \sqrt{4} = 2
\]

Thus, \( (f \circ g)(1) = 2 \).

Would you like more details or have any questions?

Here are five related questions to expand on this:
1. What is \( (g \circ f)(1) \)?
2. How would the composition change if \( g(x) = -3x + 2 \)?
3. What is the domain of \( f \circ g \)?
4. How can you visualize the composition of two functions?
5. What happens if you reverse the order of composition for \( f \) and \( g \)?

**Tip**: Always evaluate the inner function first when dealing with function compositions.

Suppose f(x) = √x^2 − 3x + 6 and g(x) = −4x + 6. Find (f ∘ g)(1).

Learn how to solve a function composition problem involving f(x) = √x^2 − 3x + 6 and g(x) = −4x + 6. This step-by-step guide walks you through finding (f ∘ g)(1), simplifying expressions with square roots, and working with linear functions. Suitable for students in Grades 9-11.

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