Question

fill out the table of values and select whether each function is odd, even or neither. for the trig functions, make sure your calculator is in radian mode. note: an expression like sin x + 5 should be typed as sin(x) + 5. write your answer to the nearest hundredth. f(x) = |x|
\begin{tabular}{|c|c|}
hline x & y \\
hline -2 & \\
hline -1 & \\
hline 0 & \\
hline 1 & \\
hline 2 & \\
hline end{tabular}

f(x) = sin x
\begin{tabular}{|c|c|}
hline x & y \\
hline -2 & \\
hline -1 & \\
hline 0 & \\
hline 1 & \\
hline 2 & \\
hline end{tabular}

f(x) = -x⁵ + 1
\begin{tabular}{|c|c|}
hline x & y \\
hline -2 & \\
hline -1 & \\
hline 0 & \\
hline 1 & \\
hline 2 & \\
hline end{tabular}

Explanation:

For \( f(x) = |x| \)

Step1: Calculate \( y \) for \( x = -2 \)

\( f(-2) = |-2| = 2 \)

Step2: Calculate \( y \) for \( x = -1 \)

\( f(-1) = |-1| = 1 \)

Step3: Calculate \( y \) for \( x = 0 \)

\( f(0) = |0| = 0 \)

Step4: Calculate \( y \) for \( x = 1 \)

\( f(1) = |1| = 1 \)

Step5: Calculate \( y \) for \( x = 2 \)

\( f(2) = |2| = 2 \)

Step6: Check parity

A function is even if \( f(-x) = f(x) \) for all \( x \). Here, \( f(-x) = |-x| = |x| = f(x) \), so it's even.

For \( f(x) = \sin x \)

Step1: Calculate \( y \) for \( x = -2 \)

\( f(-2) = \sin(-2) \approx -0.91 \) (using calculator in radian mode)

Step2: Calculate \( y \) for \( x = -1 \)

\( f(-1) = \sin(-1) \approx -0.84 \)

Step3: Calculate \( y \) for \( x = 0 \)

\( f(0) = \sin(0) = 0 \)

Step4: Calculate \( y \) for \( x = 1 \)

\( f(1) = \sin(1) \approx 0.84 \)

Step5: Calculate \( y \) for \( x = 2 \)

\( f(2) = \sin(2) \approx 0.91 \)

Step6: Check parity

A function is odd if \( f(-x) = -f(x) \). Here, \( \sin(-x) = -\sin(x) \), so it's odd.

For \( f(x) = -x^5 + 1 \)

Step1: Calculate \( y \) for \( x = -2 \)

\( f(-2) = -(-2)^5 + 1 = -(-32) + 1 = 33 \)

Step2: Calculate \( y \) for \( x = -1 \)

\( f(-1) = -(-1)^5 + 1 = -(-1) + 1 = 2 \)

Step3: Calculate \( y \) for \( x = 0 \)

\( f(0) = -0^5 + 1 = 1 \)

Step4: Calculate \( y \) for \( x = 1 \)

\( f(1) = -1^5 + 1 = -1 + 1 = 0 \)

Step5: Calculate \( y \) for \( x = 2 \)

\( f(2) = -2^5 + 1 = -32 + 1 = -31 \)

Step6: Check parity

Check \( f(-x) = -(-x)^5 + 1 = x^5 + 1 \). \( f(x) = -x^5 + 1 \). \( f(-x)
eq f(x) \) (not even) and \( f(-x)
eq -f(x) \) (since \( -f(x) = x^5 - 1
eq x^5 + 1 \)), so neither.

Table for \( f(x) = |x| \)

\( x \)	\( y \)
-1	1
0	0
1	1
2	2

Parity: Even

Table for \( f(x) = \sin x \)

\( x \)	\( y \) (approx)
-1	-0.84
0	0
1	0.84
2	0.91

Parity: Odd

Table for \( f(x) = -x^5 + 1 \)

\( x \)	\( y \)
-1	2
0	1
1	0
2	-31

Parity: Neither

Answer:

• For \( f(x) = |x| \): Table filled as above, even.
• For \( f(x) = \sin x \): Table filled as above, odd.
• For \( f(x) = -x^5 + 1 \): Table filled as above, neither.