Answer:
[tex]f(x) = 3\, x + 6[/tex].
[tex]\displaystyle f(x) = 6\, \left(\frac{3}{2}\right)^{x}[/tex].
Step-by-step explanation:
A linear function in a cartesian plane can be written in the form:
[tex]f(x) = m\, x + k[/tex],
Where [tex]m[/tex] (slope) and [tex]k[/tex] ([tex]y[/tex]-intercept) are constants.
Given the two points [tex](0,\, 6)[/tex] and [tex](1,\, 9)[/tex], obtain two equations of [tex]m[/tex] and [tex]k[/tex] by substituting in the value of [tex](x,\, f(x))[/tex] at each point:
Hence, the equation of this linear function would be:
[tex]f(x) = 3\, x + 6[/tex].
An exponential function in a cartesian plane can be written in the form:
[tex]f(x) = a\, b^{x}[/tex],
Where [tex]a \ne 0[/tex] and [tex]b > 0[/tex] are constants.
Similar to the example of the linear equation, the two points [tex](0,\, 6)[/tex] and [tex](1,\, 9)[/tex] provide two equations for [tex]a[/tex] and [tex]b[/tex]:
Hence, the equation of this exponential function would be:
[tex]\displaystyle f(x) = 6\, \left(\frac{3}{2}\right)^{x}[/tex].
