Let `f(x)={(x+1,xle4),(2x+1,4ltxle9),(-x+7,xgt9):}` and `g(x)={(x^(2),-1lexlt3),(x+2,3lexle5):}` then, find ` f(g(x))`.

Explanation

To find \( f(g(x)) \), we will follow these steps:

### Step 1: Define the Functions
We have two functions defined as follows:
- \( f(x) = \begin{cases} x + 1 & \text{if } x \leq 4 \\ 2x + 1 & \text{if } 4 < x \leq 9 \\ -x + 7 & \text{if } x > 9 \end{cases} \)

- \( g(x) = \begin{cases} x^2 & \text{if } -1 \leq x < 3 \\ x + 2 & \text{if } 3 \leq x \leq 5 \end{cases} \)

### Step 2: Determine the Range of \( g(x) \)
We need to find the range of \( g(x) \) for the two cases:

1. For \( -1 \leq x < 3 \):
- The maximum value occurs at \( x = 3 \) (not included), so \( g(x) \) approaches \( 9 \) but does not reach it.
- Therefore, \( g(x) \) ranges from \( 0 \) to \( 9 \) (not including \( 9 \)).

2. For \( 3 \leq x \leq 5 \):
- At \( x = 3 \), \( g(3) = 5 \).
- At \( x = 5 \), \( g(5) = 7 \).
- Therefore, \( g(x) \) ranges from \( 5 \) to \( 7 \).

Combining these ranges, we find that \( g(x) \) takes values in the interval \( [0, 9) \).

### Step 3: Analyze \( f(g(x)) \)
Now we will evaluate \( f(g(x)) \) based on the ranges of \( g(x) \):

1. **For \( 0 \leq g(x) \leq 4 \)**:
- Here, \( f(g(x)) = g(x) + 1 \).
- This corresponds to \( g(x) \) values from \( 0 \) to \( 4 \). Thus, for \( -1 \leq x < 2 \):
- \( f(g(x)) = g(x) + 1 = x^2 + 1 \).

2. **For \( 4 < g(x) \leq 7 \)**:
- Here, \( f(g(x)) = 2g(x) + 1 \).
- This corresponds to \( g(x) \) values from \( 5 \) to \( 7 \) (which occurs when \( 3 \leq x \leq 5 \)):
- \( f(g(x)) = 2(x + 2) + 1 = 2x + 4 + 1 = 2x + 5 \).

3. **For \( 7 < g(x) < 9 \)**:
- Here, \( f(g(x)) = -g(x) + 7 \).
- However, since \( g(x) \) does not actually reach \( 9 \) and values above \( 7 \) will not be considered as \( g(x) \) only goes up to \( 7 \).

### Step 4: Combine the Results
Now we can combine the results from the different intervals:

- For \( -1 \leq x \leq 2 \):
\[ f(g(x)) = x^2 + 1 \]

- For \( 3 \leq x \leq 5 \):
\[ f(g(x)) = 2x + 5 \]

### Final Result
Thus, we can summarize \( f(g(x)) \) as:
\[ f(g(x)) = \begin{cases} x^2 + 1 & \text{if } -1 \leq x \leq 2 \\ 2x + 5 & \text{if } 3 \leq x \leq 5 \end{cases} \]