Let g (x) = max (5 - x, x + 2). The smallest possible value of g (x) is

Options

A. `4.0`
B. 4.5
C. 1.5
D. None of these

Answer

To find the smallest possible value of the function \( g(x) = \max(5 - x, x + 2) \), we need to analyze the two expressions inside the maximum function and determine where they intersect.

### Step 1: Set the two expressions equal to each other
To find the point where the two expressions are equal, we set:
\[ 5 - x = x + 2 \]

### Step 2: Solve for \( x \)
Now, we solve the equation:
\[ 5 - x = x + 2 \]
Adding \( x \) to both sides gives:
\[ 5 = 2x + 2 \]
Subtracting \( 2 \) from both sides results in:
\[ 3 = 2x \]
Dividing both sides by \( 2 \) gives:
\[ x = \frac{3}{2} = 1.5 \]

### Step 3: Evaluate \( g(x) \) at \( x = 1.5 \)
Now we substitute \( x = 1.5 \) back into the function to find the value of \( g(1.5) \):
\[ g(1.5) = \max(5 - 1.5, 1.5 + 2) \]
Calculating each part:
\[ 5 - 1.5 = 3.5 \]
\[ 1.5 + 2 = 3.5 \]
Thus,
\[ g(1.5) = \max(3.5, 3.5) = 3.5 \]

### Step 4: Determine the behavior of \( g(x) \)
Next, we need to check the behavior of \( g(x) \) as \( x \) moves away from \( 1.5 \):
- For \( x < 1.5 \), \( 5 - x \) is greater than \( x + 2 \).
- For \( x > 1.5 \), \( x + 2 \) becomes greater than \( 5 - x \).

### Conclusion
The smallest possible value of \( g(x) \) occurs at the point where the two expressions are equal, which we found to be \( g(1.5) = 3.5 \). Therefore, the smallest possible value of \( g(x) \) is:
\[ \boxed{3.5} \]