An exponential function has the form
[tex]y=ab^x[/tex]Therefore, to find an exponential function that satisfies our condition, we need to find a and b.
From f(-2) = 1, we have
[tex]1=ab^{-2}\: ^{}\: \: \: \: \; ^{}\: \: \: \: \; (1)[/tex]and from f(7) = 63, we have
[tex]63=ab^7\: \: \: \: \; ^{}\: \: \: \: \; (2)[/tex]Solving for a in equation (1) gives
[tex]a=b^2[/tex]substituting this value of a into equation (2) gives
[tex]63=b^2\cdot b^7[/tex][tex]63=b^8[/tex][tex]\begin{gathered} \therefore b=\sqrt[8]{63} \\ b=1.6785 \end{gathered}[/tex]With the value of b in hand, we now find the value of a:
[tex]\begin{gathered} a=b^2 \\ \therefore a=2.8173 \end{gathered}[/tex]Hence, the exponential function is
[tex]f(x)=(2.8173)(1.6785)^x[/tex]Evaluating the above function at x = 1 gives
[tex]\begin{gathered} f(1)=(2.8173)(1.6785)^1 \\ \boxed{\therefore f(1)=4.73.} \end{gathered}[/tex]which is our answer!
