Questions: Suppose that the function (f) is defined, for all real numbers [ f(x)=leftbeginarrayll frac14 x+1 text if x leq-1 -(x+1)^2+1 text if -1<x<2 -frac12 x+2 text if x geq 2 endarrayright. ] Find (f(-3), f(-1)), and (f(1)). [ beginarrayl f(-3)= f(-1)= f(1)= endarray ]

$Suppose that the function (f) is defined, for all real numbers [ f(x)=leftbeginarrayll frac14 x+1 text if x leq-1 -(x+1)^2+1 text if -1<x<2 -frac12 x+2 text if x geq 2 endarrayright. ] Find (f(-3), f(-1)), and (f(1)). [ beginarrayl f(-3)= f(-1)= f(1)= endarray ]$

Transcript text: Suppose that the function $f$ is defined, for all real numbers \[ f(x)=\left\{\begin{array}{ll} \frac{1}{4} x+1 & \text { if } x \leq-1 \\ -(x+1)^{2}+1 & \text { if }-1

Solution

Solution Steps

Step 1: Evaluate \( f(-3) \)

To find \( f(-3) \), we need to determine which piece of the piecewise function applies. Since \(-3 \leq -1\), we use the first piece of the function:

\[ f(x) = \frac{1}{4}x + 1 \]

Substitute \( x = -3 \):

\[ f(-3) = \frac{1}{4}(-3) + 1 = -\frac{3}{4} + 1 = \frac{1}{4} \]

Step 2: Evaluate \( f(-1) \)

To find \( f(-1) \), we again determine which piece of the piecewise function applies. Since \(-1 \leq -1\), we use the first piece of the function:

\[ f(x) = \frac{1}{4}x + 1 \]

Substitute \( x = -1 \):

\[ f(-1) = \frac{1}{4}(-1) + 1 = -\frac{1}{4} + 1 = \frac{3}{4} \]

Step 3: Evaluate \( f(1) \)

To find \( f(1) \), we determine which piece of the piecewise function applies. Since \(-1 < 1 < 2\), we use the second piece of the function:

\[ f(x) = -(x+1)^2 + 1 \]

Substitute \( x = 1 \):

\[ f(1) = -(1+1)^2 + 1 = -(2)^2 + 1 = -4 + 1 = -3 \]

Final Answer

\[ \begin{array}{l} f(-3) = \boxed{\frac{1}{4}} \\ f(-1) = \boxed{\frac{3}{4}} \\ f(1) = \boxed{-3} \end{array} \]

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