Question Find $f(g(h(11/2)))$ given $f(x)=\sqrt{x-4}$ , $g(x)=\frac{1}{2}x+1$ , and $h(x)=2x-3$ .

Studdy Solution

STEP 1

Assumptions
1. We are given three functions: $f(x) = \sqrt{x-4}$ , $g(x) = \frac{1}{2} x + 1$ , and $h(x) = 2x - 3$ .
2. We need to find the value of the composite function $f(g(h(\frac{11}{2})))$ .
3. We will evaluate the innermost function first and proceed to the outer functions.

STEP 2

First, we need to evaluate $h(x)$ at $x = \frac{11}{2}$ .
$h\left(\frac{11}{2}\right) = 2 \cdot \frac{11}{2} - 3$

STEP 3

Simplify the expression by multiplying 2 with $\frac{11}{2}$ .
$h\left(\frac{11}{2}\right) = 11 - 3$

STEP 4

Subtract 3 from 11 to find the value of $h\left(\frac{11}{2}\right)$ .
$h\left(\frac{11}{2}\right) = 8$

STEP 5

Next, we need to evaluate $g(x)$ at $x = h\left(\frac{11}{2}\right)$ , which we found to be 8.
$g(8) = \frac{1}{2} \cdot 8 + 1$

STEP 6

Multiply $\frac{1}{2}$ by 8.
$g(8) = 4 + 1$

STEP 7

Add 1 to 4 to find the value of $g(8)$ .
$g(8) = 5$

STEP 8

Now, we need to evaluate $f(x)$ at $x = g(8)$ , which we found to be 5.
$f(5) = \sqrt{5 - 4}$

STEP 9

Subtract 4 from 5.
$f(5) = \sqrt{1}$

STEP 10

Calculate the square root of 1.
$f(5) = 1$

STEP 11

The value of the composite function $f(g(h(\frac{11}{2})))$ is 1.
The answer is 1.

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