an exponential function ( f(x)=a b^{x} ) passes through the points ( (0,9) ) and ( (3,576) ). what are the…

Explanation:

Step1: Substitute the point ((0,9)) into the function

Substitute (x = 0) and (f(x)=9) into (f(x)=ab^{x}). We get (9 = ab^{0}). Since (b^{0}=1) for (b\neq0), then (a = 9).

Step2: Substitute (a = 9) and the point ((3,576)) into the function

The function becomes (f(x)=9b^{x}). Substitute (x = 3) and (f(x)=576) into (f(x)=9b^{x}), we have (576=9b^{3}). First, divide both sides by (9): (\frac{576}{9}=b^{3}), so (64 = b^{3}). Then, take the cube - root of both sides: (b=\sqrt[3]{64}). Since (4\times4\times4 = 64), (b = 4).

Answer:

(a = 9), (b = 4)