The range of `4+5` cos x is

Options

A. `[-1, 9]`
B. `(-1, 9]`
C. `(-1, 9)`
D. `[-1, 9)`

Answer

To find the range of the expression \(4 + 5 \cos x\), we can follow these steps:

### Step 1: Determine the range of \(\cos x\)
The cosine function, \(\cos x\), has a range of \([-1, 1]\). This means that for any value of \(x\), \(\cos x\) will always be between -1 and 1.

### Step 2: Multiply the range of \(\cos x\) by 5
Next, we multiply the entire range of \(\cos x\) by 5:
\[ 5 \cdot (-1) \leq 5 \cos x \leq 5 \cdot 1 \]
This simplifies to:
\[ -5 \leq 5 \cos x \leq 5 \]

### Step 3: Add 4 to the entire inequality
Now, we add 4 to each part of the inequality:
\[ -5 + 4 \leq 4 + 5 \cos x \leq 5 + 4 \]
This simplifies to:
\[ -1 \leq 4 + 5 \cos x \leq 9 \]

### Step 4: State the range
From the above inequality, we can conclude that the range of \(4 + 5 \cos x\) is:
\[ [-1, 9] \]

### Final Answer
Thus, the range of \(4 + 5 \cos x\) is \([-1, 9]\).

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