PieceWise Functions

To determine the number of values of \( x \) for which \( f(f(x)) = 5 \), we need to analyze the given piecewise function:

 

\[
f(x) = \begin{cases} 
x^2 - 4 & \text{if } x \geq -4 \\
x + 3 & \text{otherwise}
\end{cases}
\]

 

We'll consider each case separately.

 

### Case 1: \( x \geq -4 \)

 

For \( x \geq -4 \), \( f(x) = x^2 - 4 \). We need to find \( y \) such that:

 

\[
f(y) = 5
\]

 

So,

 

\[
y^2 - 4 = 5 \implies y^2 = 9 \implies y = \pm 3
\]

 

However, since \( x \geq -4 \), both solutions \( y = 3 \) and \( y = -3 \) are valid because they are within the domain \( x \geq -4 \). Hence, \( y = 3 \) and \( y = -3 \).

 

Next, we need \( f(x) \) such that:

 

\[
f(x) = 3 \quad \text{or} \quad f(x) = -3
\]

 

#### Subcase 1.1: \( f(x) = 3 \)

 

\[
x^2 - 4 = 3 \implies x^2 = 7 \implies x = \pm \sqrt{7}
\]

 

Since \( x \geq -4 \), both solutions \( x = \sqrt{7} \) and \( x = -\sqrt{7} \) are valid.

 

#### Subcase 1.2: \( f(x) = -3 \)

 

\[
x^2 - 4 = -3 \implies x^2 = 1 \implies x = \pm 1
\]

 

Both solutions \( x = 1 \) and \( x = -1 \) are valid since \( x \geq -4 \).

 

### Case 2: \( x < -4 \)

 

For \( x < -4 \), \( f(x) = x + 3 \). We need to find \( y \) such that:

\[
f(y) = 5
\]

 

So,

 

\[
y + 3 = 5 \implies y = 2
\]

 

However, since \( x < -4 \), the solution \( y = 2 \) does not fall within this domain. Therefore, there are no solutions from this case.

### Conclusion

 

Summarizing the valid solutions from both subcases under \( x \geq -4 \), we have:

 

- \( f(x) = 3 \): \( x = \sqrt{7}, -\sqrt{7} \)
- \( f(x) = -3 \): \( x = 1, -1 \)

 

Thus, we find a total of \( 4 \) values of \( x \):

 

\[
x = \sqrt{7}, -\sqrt{7}, 1, -1
\]

 

Therefore, the number of values of \( x \) for which \( f(f(x)) = 5 \) is \( \boxed{4} \).