If f(x) = 2x - 1 and g(x) = 4x + 5 , find the following. (a) f(g()) = (b) g(f()) = () f(f(x)) =

Let's solve the given problem step by step.

We are given two functions:

- \( f(x) = 2x - 1 \)
- \( g(x) = 4x + 5 \)

### Part (a) \( f(g(x)) \)
This means we need to substitute \( g(x) \) into \( f(x) \):

\[
f(g(x)) = f(4x + 5)
\]

Substitute \( 4x + 5 \) into \( f(x) = 2x - 1 \):

\[
f(4x + 5) = 2(4x + 5) - 1 = 8x + 10 - 1 = 8x + 9
\]

So, \( f(g(x)) = 8x + 9 \).

---

### Part (b) \( g(f(x)) \)
This means we need to substitute \( f(x) \) into \( g(x) \):

\[
g(f(x)) = g(2x - 1)
\]

Substitute \( 2x - 1 \) into \( g(x) = 4x + 5 \):

\[
g(2x - 1) = 4(2x - 1) + 5 = 8x - 4 + 5 = 8x + 1
\]

So, \( g(f(x)) = 8x + 1 \).

---

### Part (c) \( f(f(x)) \)
This means we need to substitute \( f(x) \) into \( f(x) \):

\[
f(f(x)) = f(2x - 1)
\]

Substitute \( 2x - 1 \) into \( f(x) = 2x - 1 \):

\[
f(2x - 1) = 2(2x - 1) - 1 = 4x - 2 - 1 = 4x - 3
\]

So, \( f(f(x)) = 4x - 3 \).

---

### Final Answers:
- \( f(g(x)) = 8x + 9 \)
- \( g(f(x)) = 8x + 1 \)
- \( f(f(x)) = 4x - 3 \)

Would you like to dive deeper into any of these steps?

Here are five related questions to expand on these concepts:

1. What happens if we switch the order of composition for any two arbitrary functions?
2. Can you find \( g(g(x)) \) using the same method?
3. How do we prove that function composition is associative?
4. Can you solve \( f(g(x)) = 0 \) for \( x \)?
5. What is the graphical interpretation of function composition?

**Tip**: When composing functions, always substitute the inner function into the outer function carefully by replacing every occurrence of the variable.

If f(x) = 2x - 1 and g(x) = 4x + 5, find the following: (a) f(g(x)) = ? (b) g(f(x)) = ? (c) f(f(x)) = ?

Explore the composition of two functions f(x) = 2x - 1 and g(x) = 4x + 5 with a detailed step-by-step solution. Learn how to find f(g(x)), g(f(x)), and f(f(x)) using algebraic techniques. Suitable for students in Grades 9-11.

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