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).

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 $-2$.

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

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

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

Final Answer