Compute different values for y using the given equation and some values for x. Then, plot the corresponding points on the coordinate plane, draw a graph, and determine the domain and range from the graph.
The given equation is:
[tex]y=(\frac{3}{8})^x[/tex]
For the value x=-2:
[tex]\begin{gathered} y=(\frac{3}{8})^{-2} \\ =(\frac{8}{3})^2 \\ =\frac{8^2}{3^2} \\ =\frac{64}{9} \\ \approx7.1 \end{gathered}[/tex]
Compute the value for x=-1, x=0 and x=1:
[tex]\begin{gathered} x=-1 \\ \Rightarrow y=(\frac{3}{8})^{-1} \\ =\frac{8}{3} \\ \approx2.7 \end{gathered}[/tex][tex]\begin{gathered} x=0 \\ \Rightarrow y=(\frac{3}{8})^0 \\ =1 \end{gathered}[/tex][tex]\begin{gathered} x=1 \\ \Rightarrow y=(\frac{3}{8})^1 \\ =\frac{3}{8} \\ \approx0.38 \end{gathered}[/tex]
Plot the following points on a coordinate plane:
[tex]\begin{gathered} (-2,7.1) \\ (-1,2.7) \\ (0,1) \\ (1,0.38) \end{gathered}[/tex]
Next, draw a line through those points:
The domain of the function is the set of all the values of x that the expression can take as an input. Since the expression does not have any restrictions over the variable x (such as the variable x being on the denominator or inside an even root), then x can take any value.
Then, the domain is all the real numbers.
The range of the function is the set of all values that the variable y can take when the expression is evaluated for values of x in the domain. In this case, there is no value of x such that y is less than or equal to 0, which we can see on the graph. On the other hand, y can take values as near to 0 as we want, or as big as we want.
Then, the range is the set of all positive numbers.
