For the following​ function, find the value of​ (a) f left parenthesis negative 4 right parenthesis and​ (b) f left parenthesis 2 right parenthesis​, if possible.

Question

y equals left brace Start 2 By 2 Matrix 1st Row 1st Column x squared minus 2 2nd Column if x less than or equals 0 Superscript 2nd Row 1st Column x cubed plus 3 2nd Column if x greater than 0 Superscript EndMatrix

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Part 1
​(a) Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your answer.
A.
f left parenthesis negative 4 right parenthesisequals
  
enter your response here ​(Simplify your​ answer.)
B.
There is no solution.

Answer 1

To evaluate the function \( f(x) \) defined as:

\[ y = \begin{cases} x^2 - 2 & \text{if } x \leq 0 \ x^3 + 3 & \text{if } x > 0 \end{cases} \]

we will find the values for \( f(-4) \) and \( f(2) \).

Part (a): Finding \( f(-4) \)

Since -4 is less than 0, we will use the first case of the piecewise function:

\[ f(-4) = (-4)^2 - 2 \]

Calculating this:

\[ = 16 - 2 = 14 \]

So,

\[ f(-4) = 14 \]

Part (b): Finding \( f(2) \)

Since 2 is greater than 0, we will use the second case of the piecewise function:

\[ f(2) = 2^3 + 3 \]

Calculating this:

\[ = 8 + 3 = 11 \]

So,

\[ f(2) = 11 \]

Summary of Results:

(a) \( f(-4) = 14 \)
(b) \( f(2) = 11 \)