Evaluate each of the following expressions given the function definitions for f, g, and h.
f
(
x
)
=
√
x
+
5

g
(
x
)
=
4
x
−
8

h
(
x
)
=
7
x

f
(
g
(
5
)
)
=
   

g
(
f
(
87
)
)
=
   

h
(
g
(
f
(
2
)
)
)
=
   

Let's evaluate each expression step-by-step.

### Given Function Definitions:
1. \( f(x) = \sqrt{x} + 5 \)
2. \( g(x) = 4x - 8 \)
3. \( h(x) = 7x \)

Now, let's evaluate each expression in sequence.

---

#### Expression 1: \( f(g(5)) \)
1. **Evaluate \( g(5) \)**:
   \[
   g(5) = 4(5) - 8 = 20 - 8 = 12
   \]

2. **Substitute into \( f(x) \)**:
   \[
   f(g(5)) = f(12) = \sqrt{12} + 5
   \]

   Simplify:
   \[
   f(12) = \sqrt{12} + 5 \approx 3.464 + 5 = 8.464
   \]

So, \( f(g(5)) \approx 8.464 \).

---

#### Expression 2: \( g(f(87)) \)
1. **Evaluate \( f(87) \)**:
   \[
   f(87) = \sqrt{87} + 5
   \]

   Approximate \( \sqrt{87} \):
   \[
   f(87) \approx 9.327 + 5 = 14.327
   \]

2. **Substitute into \( g(x) \)**:
   \[
   g(f(87)) = g(14.327) = 4(14.327) - 8
   \]

   Multiply and simplify:
   \[
   g(14.327) \approx 57.308 - 8 = 49.308
   \]

So, \( g(f(87)) \approx 49.308 \).

---

#### Expression 3: \( h(g(f(2))) \)
1. **Evaluate \( f(2) \)**:
   \[
   f(2) = \sqrt{2} + 5
   \]

   Approximate \( \sqrt{2} \):
   \[
   f(2) \approx 1.414 + 5 = 6.414
   \]

2. **Substitute into \( g(x) \)**:
   \[
   g(f(2)) = g(6.414) = 4(6.414) - 8
   \]

   Multiply and simplify:
   \[
   g(6.414) \approx 25.656 - 8 = 17.656
   \]

3. **Substitute into \( h(x) \)**:
   \[
   h(g(f(2))) = h(17.656) = 7(17.656)
   \]

   Multiply:
   \[
   h(17.656) \approx 123.592
   \]

So, \( h(g(f(2))) \approx 123.592 \).

---

### Final Answers:
1. \( f(g(5)) \approx 8.464 \)
2. \( g(f(87)) \approx 49.308 \)
3. \( h(g(f(2))) \approx 123.592 \)

Would you like further details on any step?

---

Here are five related questions to extend your understanding:

1. What happens if we replace \( f(x) \) with \( f(x) = x^2 + 5 \) and re-evaluate these expressions?
2. How would the answers change if we used \( g(x) = x^3 - 8 \) instead?
3. If \( h(x) = 2x + 3 \), what would be the result of \( h(g(f(5))) \)?
4. Can we find \( f(g(x)) \) as a function of \( x \) instead of specific values?
5. How can these compositions of functions be visualized graphically?

**Tip:** When working with function compositions, breaking down each step carefully helps prevent errors and simplifies complex expressions.

Evaluate each of the following expressions given the function definitions for f, g, and h.
f(x) = √x + 5
g(x) = 4x - 8
h(x) = 7x

f(g(5)) = ?
g(f(87)) = ?
h(g(f(2))) = ?

Solve function composition problems involving f(x) = √x + 5, g(x) = 4x - 8, and h(x) = 7x. Learn step-by-step how to evaluate f(g(5)), g(f(87)), and h(g(f(2))). This solution includes algebraic simplifications and square root approximations. Suitable for Grades 9-11.

Math Problem Statement

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

g ( x )

h ( x )

f ( g ( 5 ) )

g ( f ( 87 ) )

h ( g ( f ( 2 ) ) )

Solution

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