Find the range of each of the following functions: (where {.} and [.] represents fractional part and greatest integer part functions respectively)
`f(x)=5+3sinx+4cosx`

Explanation

To find the range of the function \( f(x) = 5 + 3 \sin x + 4 \cos x \), we can follow these steps:

### Step 1: Identify the coefficients
We have the function in the form \( f(x) = 5 + 3 \sin x + 4 \cos x \). Here, the coefficients of sine and cosine are \( a = 3 \) and \( b = 4 \).

### Step 2: Determine the maximum and minimum values of \( 3 \sin x + 4 \cos x \)
The expression \( 3 \sin x + 4 \cos x \) can be analyzed using the formula for the range of \( a \sin x + b \cos x \), which is given by:
\[ -\sqrt{a^2 + b^2} \leq a \sin x + b \cos x \leq \sqrt{a^2 + b^2} \]
Calculating \( \sqrt{a^2 + b^2} \):
\[ \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]
Thus, we have:
\[ -5 \leq 3 \sin x + 4 \cos x \leq 5 \]

### Step 3: Add 5 to the entire inequality
Now, we will add 5 to each part of the inequality to find the range of \( f(x) \):
\[ -5 + 5 \leq 3 \sin x + 4 \cos x + 5 \leq 5 + 5 \]
This simplifies to:
\[ 0 \leq f(x) \leq 10 \]

### Step 4: State the range of the function
Thus, the range of the function \( f(x) = 5 + 3 \sin x + 4 \cos x \) is:
\[ [0, 10] \]

### Final Answer
The range of \( f(x) \) is \( [0, 10] \).

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