When a function is plotted on a graph, the domain and the range of the function are the x-coordinate and the y-coordinate respectively.
The domain and the range of the given function are:
Domain: [tex](-\infty,\infty)[/tex]
Range: [tex](3,\infty)[/tex]
The given function is:
[tex]g(x) = 3^x + 3[/tex]
First, we plot the graph of g(x)
To do this, we need to generate values for x and g(x). The table is generated as follows:
[tex]x = 0 \to g(0) = 3^0 + 3 = 4[/tex]
[tex]x = 1 \to g(1) = 3^1 + 3 = 6[/tex]
[tex]x = 2 \to g(2) = 3^2 + 3 = 12[/tex]
[tex]x = 3 \to g(3) = 3^3 + 3 = 30[/tex]
[tex]x = 4 \to g(4) = 3^4 + 3 = 84[/tex]
The generated values in tabular form are:
[tex]\begin{array}{cccccc}x & {0} & {1} & {2} & {3} & {4} \ \\ g(x) & {4} & {6} & {12} & {30} & {84} \ \end{array}[/tex]
Refer to the attached image for graph of g(x)
To determine the domain, we simply observe the x-axis.
The curve stretches through the x-axis, and there are no visible endpoints on the axis. This means that the curve starts from [tex]-\infty[/tex] to [tex]+\infty[/tex]
Hence, the domain of the function is: [tex](-\infty,\infty)[/tex]
To determine the range, we simply observe the y-axis.
The curve of g(x) starts at y = 3 on the y-axis and the curve faces upward direction. This means that the curve of g(x) is greater than 3 on the y-axis.
Hence, the range of the function is: [tex](3,\infty)[/tex]
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