To determine which statement about the piecewise function \( f(x) \) is true, we need to evaluate \( f(-1) \) and \( f(1) \) based on the defined intervals of the function.
The piecewise function is defined as follows:
- \( f(x) = (x + 3)^2 - 1 \) for \( -5 \leq x \leq -1 \)
- \( f(x) = -x \) for \( -1 < x \leq 1 \)
- \( f(x) = \frac{5}{2} \log_2(-x + 4) - 1 \) for \( 1 < x \leq 4 \)
Now, let's evaluate \( f(-1) \) and \( f(1) \):
Evaluate \( f(-1) \):
Since \( -1 \) falls on the boundary of the first interval, we will use the first piece: \[ f(-1) = (-1 + 3)^2 - 1 = (2)^2 - 1 = 4 - 1 = 3 \]
Evaluate \( f(1) \):
Since \( 1 \) also falls on the boundary of the second interval, we use the second piece: \[ f(1) = -1 \]
Compare \( f(-1) \) and \( f(1) \):
Now we compare the two values we calculated: \[ f(-1) = 3 \quad \text{and} \quad f(1) = -1 \] Thus, \( f(-1) > f(1) \).
Conclusion:
The correct statement is: \[ f(-1) > f(1) \]
So the answer is:
- \( f(-1) > f(1) \)
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