Solved on Dec 10, 2023

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

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

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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Find f(g(h(11/2)))f(g(h(11/2)))f(g(h(11/2))) given f(x)=x−4f(x)=\sqrt{x-4}f(x)=x−4​, g(x)=12x+1g(x)=\frac{1}{2}x+1g(x)=21​x+1, and h(x)=2x−3h(x)=2x-3h(x)=2x−3.

STEP 1

STEP 2

First, we need to evaluate h(x) h(x) h(x) at x=112 x = \frac{11}{2} x=211​.h(112)=2⋅112−3 h\left(\frac{11}{2}\right) = 2 \cdot \frac{11}{2} - 3 h(211​)=2⋅211​−3

STEP 3

Simplify the expression by multiplying 2 with 112 \frac{11}{2} 211​.h(112)=11−3 h\left(\frac{11}{2}\right) = 11 - 3 h(211​)=11−3

STEP 4

Subtract 3 from 11 to find the value of h(112) h\left(\frac{11}{2}\right) h(211​).h(112)=8 h\left(\frac{11}{2}\right) = 8 h(211​)=8

STEP 5

Next, we need to evaluate g(x) g(x) g(x) at x=h(112) x = h\left(\frac{11}{2}\right) x=h(211​), which we found to be 8.g(8)=12⋅8+1 g(8) = \frac{1}{2} \cdot 8 + 1 g(8)=21​⋅8+1

STEP 6

Multiply 12 \frac{1}{2} 21​ by 8.g(8)=4+1 g(8) = 4 + 1 g(8)=4+1

STEP 7

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

STEP 8

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

STEP 9

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

STEP 10

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

STEP 11

The value of the composite function f(g(h(112))) f(g(h(\frac{11}{2}))) f(g(h(211​))) is 1.The answer is 1.

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

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

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

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

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

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

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

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

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

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

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