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Step-by-step Answer
PAGE: 166SET: ExercisesPROBLEM: 1
Please look in your text book for this problem Statement

The function is f(x)=2^{-x}.

Make the table of values to find ordered pairs that satisfy the function.

Choose values for x and find the corresponding values for y

x f(x)=2^{-x} (x, y)
-4 f(-4)=2^{-(-4)}=16 (-4, 16)
-3 f(-3)=2^{-(-3)}=8 (-3, 8)
-2 f(-2)=2^{-(-2)}=4 (-2, 4)
0 f(0)=2^{0}=1 (0, 1)
2 f(2)=2^{-(2)}=0.25 (2, 0.25)
4 f(4)=2^{-(4)}=0.06 (4, 0.06)
6 f(6)=2^{-(6)}=0.01 (6, 0.01)
8 f(8)=2^{-(8)}=0.003 (8, 0.003)

Graph:

1. Draw a coordinate plane.

2. Plot the coordinate points.

3. Then sketch the graph, connecting the points with a smooth curve.

Observe the above graph :

Domain of the function is all real numbers.

Range of the function is (0, \infty ).

The function does not have the x-intercept.

y - intercepts is 1.

The line y=L. is the horizontal asymptote of the curve y=f(x).

if either \lim_{x \to \infty}f(x)=L or \lim_{x \to -\infty}f(x)=L.

\lim_{x \to \infty}2^{-x}

\\\lim_{x \to \infty}\frac{1}{2^{x}}\\\\=\frac{1}{2^{\infty}}\\\\

=0.

y=0 is the horizontal asymptote of the function.

Observe the above graph the function does not have vertical asymptote.

End behavior : \lim_{x\rightarrow -\infty }(f(x))=\infty and \lim_{x\rightarrow \infty }f(x)=0.

The function is continuous for all real numbers.

Decreasing on the interval : \left (-\infty \right, \infty ).

The graph of the function f(x)=2^{-x} is :

Domain of the function is all real numbers.

Range of the function is (0, \infty ).

The function does not have any x -intercept.

y - intercepts is 0

Horizontal asymptote of the function is y=0.

End behavior : \lim_{x\rightarrow -\infty }(f(x))=\infty and \lim_{x\rightarrow \infty }f(x)=0.

The function is decreasing over the interval : \left (-\infty \right, \infty ).



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