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) = \tan x\\) \\(\

$$\begin{array}{|c|c|} \\hline x & y \\\\ \\hline -2 & \\\\ \\hline -1 & \\\\ \\hline 0 & \\\\ \\hline 1 & \\\\ \\hline 2 & \\\\ \\hline \\end{array}$$

\\) \\(f(x) = \sin x - 3\\) \\(\

$$\begin{array}{|c|c|} \\hline x & y \\\\ \\hline -2 & \\\\ \\hline -1 & \\\\ \\hline 0 & \\\\ \\hline 1 & \\\\ \\hline 2 & \\\\ \\hline \\end{array}$$

\\) \\(f(x) = |x| - 4\\) \\(\

$$\begin{array}{|c|c|} \\hline x & y \\\\ \\hline -2 & \\\\ \\hline -1 & \\\\ \\hline 0 & \\\\ \\hline 1 & \\\\ \\hline 2 & \\\\ \\hline \\end{array}$$

\\)

Explanation:

For \( f(x)=\tan x \)

Step1: Calculate \( f(-2) \)

Using calculator (radian mode), \( \tan(-2) \approx -2.18 \)

Step2: Calculate \( f(-1) \)

\( \tan(-1) \approx -1.56 \)

Step3: Calculate \( f(0) \)

\( \tan(0) = 0 \)

Step4: Calculate \( f(1) \)

\( \tan(1) \approx 1.56 \)

Step5: Calculate \( f(2) \)

\( \tan(2) \approx 2.18 \)

Step6: Check odd/even

A function is odd if \( f(-x)=-f(x) \). Here, \( f(-2)=-f(2) \), \( f(-1)=-f(1) \), \( f(0)=-f(0)=0 \). So it's odd.

For \( f(x)=\sin x - 3 \)

Step1: Calculate \( f(-2) \)

\( \sin(-2)-3 \approx -0.91 - 3=-3.91 \)

Step2: Calculate \( f(-1) \)

\( \sin(-1)-3 \approx -0.84 - 3=-3.84 \)

Step3: Calculate \( f(0) \)

\( \sin(0)-3 = 0 - 3=-3 \)

Step4: Calculate \( f(1) \)

\( \sin(1)-3 \approx 0.84 - 3=-2.16 \) (Wait, correction: \( \sin(1)\approx0.8415 \), so \( 0.8415 - 3\approx -2.16 \)? No, wait \( \sin(1)\approx0.84 \), so \( 0.84 - 3=-2.16 \)? Wait no, \( \sin(1)\approx0.8415 \), so \( 0.8415 - 3 = -2.1585\approx -2.16 \)? Wait no, earlier \( f(-1)=\sin(-1)-3=-\sin(1)-3\approx -0.8415 - 3=-3.8415\approx -3.84 \), \( f(1)=\sin(1)-3\approx0.8415 - 3=-2.1585\approx -2.16 \). Then \( f(-2)=\sin(-2)-3=-\sin(2)-3\approx -0.9093 - 3=-3.9093\approx -3.91 \), \( f(2)=\sin(2)-3\approx0.9093 - 3=-2.0907\approx -2.09 \). Wait, my mistake earlier. Let's recalculate:

Step1 (corrected): \( f(-2)=\sin(-2)-3 = -\sin(2)-3 \approx -0.9093 - 3 = -3.9093 \approx -3.91 \)

Step2 (corrected): \( f(-1)=\sin(-1)-3 = -\sin(1)-3 \approx -0.8415 - 3 = -3.8415 \approx -3.84 \)

Step3: \( f(0)=\sin(0)-3 = 0 - 3 = -3 \)

Step4 (corrected): \( f(1)=\sin(1)-3 \approx 0.8415 - 3 = -2.1585 \approx -2.16 \)

Step5 (corrected): \( f(2)=\sin(2)-3 \approx 0.9093 - 3 = -2.0907 \approx -2.09 \)

Step6: Check odd/even

\( f(-x)=-\sin(x)-3 \), \( -f(x)=-\sin(x)+3 \). Since \( f(-x)
eq f(x) \) and \( f(-x)
eq -f(x) \), it's neither? Wait no, wait \( f(-x)=\sin(-x)-3 = -\sin(x)-3 \), \( -f(x)=-\sin(x)+3 \). So \( f(-x) \) is not equal to \( f(x) \) (which would be \( \sin(x)-3 \)) and not equal to \( -f(x) \). So neither. Wait but wait, maybe I miscalculated \( f(1) \) and \( f(2) \) earlier. Let's use more precise values:

\( \sin(1)\approx0.8414709848 \), so \( f(1)=0.8414709848 - 3 = -2.158529015 \approx -2.16 \)

\( \sin(2)\approx0.9092974268 \), so \( f(2)=0.9092974268 - 3 = -2.090702573 \approx -2.09 \)

\( \sin(-1)=-0.8414709848 \), so \( f(-1)=-0.8414709848 - 3 = -3.841470985 \approx -3.84 \)

\( \sin(-2)=-0.9092974268 \), so \( f(-2)=-0.9092974268 - 3 = -3.909297427 \approx -3.91 \)

Now, check \( f(-x) + f(x) = (-\sin x - 3) + (\sin x - 3) = -6
eq 0 \), and \( f(-x) - (-f(x)) = (-\sin x - 3) - (-\sin x + 3) = -6
eq 0 \). So it's neither.

For \( f(x)=|x| - 4 \)

Step1: Calculate \( f(-2) \)

\( |-2| - 4 = 2 - 4 = -2 \)

Step2: Calculate \( f(-1) \)

\( |-1| - 4 = 1 - 4 = -3 \)

Step3: Calculate \( f(0) \)

\( |0| - 4 = 0 - 4 = -4 \)

Step4: Calculate \( f(1) \)

\( |1| - 4 = 1 - 4 = -3 \)

Step5: Calculate \( f(2) \)

\( |2| - 4 = 2 - 4 = -2 \)

Step6: Check odd/even

A function is even if \( f(-x)=f(x) \). Here, \( f(-2)=f(2)=-2 \), \( f(-1)=f(1)=-3 \), \( f(0)=f(0)=-4 \). So it's even.

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

\( x \)	\( y \)
-1	-1.56
0	0
1	1.56
2	2.18

(Function: Odd)

Table for \( f(x)=\sin x - 3 \)

\( x \)	\( y \)
-1	-3.84
0	-3
1	-2.16
2	-2.09

(Function: Neither)

Table for \( f(x)=|x| - 4 \)

\( x \)	\( y \)
-1	-3
0	-4
1	-3
2	-2

(Function: Even)

Answer:

(Tables and Odd/Even):

• For \( f(x)=\tan x \): Table as above, Odd.
• For \( f(x)=\sin x - 3 \): Table as above, Neither.
• For \( f(x)=|x| - 4 \): Table as above, Even.