Evaluate each of the following expressions given the function definitions for 
f
, 
g
, and 
h
.

f
(
x
)
=
√
x
+
4

g
(
x
)
=
3
x
−
8

h
(
x
)
=
7
x

f
(
g
(
4
)
)
=
Incorrect    

g
(
f
(
85
)
)
=
Incorrect    

h
(
g
(
f
(
2
)
)
)
=
Incorrect    


Let's evaluate each of the given expressions step-by-step using the provided function definitions:

- \( f(x) = \sqrt{x} + 4 \)
- \( g(x) = 3x - 8 \)
- \( h(x) = 7x \)

### 1. \( f(g(4)) \)

First, we need to compute \( g(4) \):
\[
g(4) = 3(4) - 8 = 12 - 8 = 4
\]

Now, substitute \( g(4) = 4 \) into \( f(x) \):
\[
f(4) = \sqrt{4} + 4 = 2 + 4 = 6
\]

So, \( f(g(4)) = 6 \).

### 2. \( g(f(85)) \)

First, compute \( f(85) \):
\[
f(85) = \sqrt{85} + 4 \approx 9.22 + 4 = 13.22
\]

Now, substitute \( f(85) \approx 13.22 \) into \( g(x) \):
\[
g(13.22) = 3(13.22) - 8 = 39.66 - 8 = 31.66
\]

So, \( g(f(85)) \approx 31.66 \).

### 3. \( h(g(f(2))) \)

First, compute \( f(2) \):
\[
f(2) = \sqrt{2} + 4 \approx 1.41 + 4 = 5.41
\]

Now, substitute \( f(2) \approx 5.41 \) into \( g(x) \):
\[
g(5.41) = 3(5.41) - 8 = 16.23 - 8 = 8.23
\]

Finally, substitute \( g(f(2)) \approx 8.23 \) into \( h(x) \):
\[
h(8.23) = 7(8.23) = 57.61
\]

So, \( h(g(f(2))) \approx 57.61 \).

### Final Results:

- \( f(g(4)) = 6 \)
- \( g(f(85)) \approx 31.66 \)
- \( h(g(f(2))) \approx 57.61 \)

Would you like further details or have any questions?

Here are five related questions to expand on this:
1. What is the domain of the function \( f(x) \)?
2. How do function compositions affect the domain of the resulting function?
3. Can you compute \( f(g(10)) \) using the same method?
4. What happens if you reverse the composition, such as \( g(h(x)) \)?
5. What are the practical applications of function compositions?

**Tip:** Always check if the input to a function lies within its domain before computing its value!

Evaluate each of the following expressions given the function definitions for f, g, and h: f(x) = √x + 4, g(x) = 3x - 8, h(x) = 7x. f(g(4)) = Incorrect, g(f(85)) = Incorrect, h(g(f(2))) = Incorrect

Learn how to evaluate function compositions such as f(g(4)), g(f(85)), and h(g(f(2))) using square root, linear, and multiplication functions. Step-by-step solutions are provided for each expression. Ideal for students in Grades 9-12 studying algebra.

Math Problem Statement

f ( x )

g ( x )

h ( x )

f ( g ( 4 ) )

g ( f ( 85 ) )

h ( g ( f ( 2 ) ) )

Solution

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