One of the best ways to solve Domain-Range problems, is to draw the graph of the function.
All graphs of the form [tex]f(x)=a^{x} [/tex], where a>1,
like [tex]f(x)=2^{x} [/tex], [tex]f(x)=3^{x} [/tex], [tex]f(x)=5.1^{x} [/tex] etc, have the following properties:
i) the graph is increasing,
ii) the graph is above the x-axis, with x-axis as a horizontal asymptote,
that is the graph gets very very close to the x-axis but never touches it.
iii) the intersection of the graph and the y-axis is the point (0, 1).
So, the graph of any function [tex]f(x)=a^{x} [/tex], where a>1, looks like the graph in picture 1.
Remark
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if we go toward the negative x-axis, that is if we take smaller and smaller values of x, [tex]y=3^{x} [/tex] becomes smaller and smaller.
for example for x=-100, [tex]y=3^{-100}= \frac{1}{3^{100} } [/tex]
for x=-1000 , [tex]y=3^{-1000}= \frac{1}{3^{1000} } [/tex]
and so on...
for smaller and smaller values of x, the values of y are smaller and smaller, that is the graph is lower and lower,
becoming very very small, almost 0, but never 0.
this explains both the increasing nature of the graph (as we go towards the positive x-axis) and the asymptote line y=0, that is the x-axis.
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From the graph we see that the Domain of [tex]f(x)=3^{x} [/tex] is all Real numbers, as x can be any number,
and the Range is (0, +∞), as the function can take any value larger than 0.
The graph of [tex]f(x)=3^{x} +5 [/tex] is the graph of [tex]f(x)=3^{x} [/tex] shifted 5 units up, as shown in figure 2.
The asymptote y=0, is shifted 5 units up, to y=5.
Thus the Range becomes (5, +∞), while the Domain is still all Real numbers.
Answer:
B) Domain (-∞, ∞), Range (5, ∞)
