Questions: Suppose that the function f is defined, for all real numbers, as follows. f(x) = 1/2 x^2 - 5 if x ≠ -2 -2 if x = -2 Find f(-5), f(-2), and f(3).

Transcript text: Suppose that the function $f$ is defined, for all real numbers, as follows. \[ f(x)=\left\{\begin{array}{ll} \frac{1}{2} x^{2}-5 & \text { if } x \neq-2 \\ -2 & \text { if } x=-2 \end{array}\right. \] Find $f(-5), f(-2)$, and $f(3)$

Solution

Solution Steps

To find the values of the function $ f(x) $ at specific points, we need to evaluate the function based on the given piecewise definition. For $ x \neq -2 $, we use the expression $ \frac{1}{2} x^2 - 5 $. For $ x = -2 $, we use the value $ -2 $.

Step 1: Evaluate $ f(-5) $

For $ x = -5 $, since $ x \neq -2 $, we use the expression $ f(x) = \frac{1}{2} x^2 - 5 $: \[ f(-5) = \frac{1}{2} (-5)^2 - 5 = \frac{1}{2} \cdot 25 - 5 = 12.5 - 5 = 7.5 \]

Step 2: Evaluate $ f(-2) $

For $ x = -2 $, we use the value given directly in the piecewise function: \[ f(-2) = -2 \]

Step 3: Evaluate $ f(3) $

For $ x = 3 $, since $ x \neq -2 $, we use the expression $ f(x) = \frac{1}{2} x^2 - 5 $: \[ f(3) = \frac{1}{2} (3)^2 - 5 = \frac{1}{2} \cdot 9 - 5 = 4.5 - 5 = -0.5 \]