Ryan decides to find the exponential function that passes through the points \( (3,135) \) and \( (1,60) \). Use these points to write the equation of his function of the form \( f(x)=a b^{x} \).

To find the exponential function of the form \( f(x) = a b^{x} \) that passes through the points \( (3,135) \) and \( (1,60) \), we can set up a system of equations using these points. By substituting the coordinates into the equation, we have: 1. For point \( (3,135) \): \[ 135 = a b^3 \quad (1) \] 2. For point \( (1,60) \): \[ 60 = a b^1 \quad (2) \] Now, we can solve these equations. From equation (2), we can express \( a \): \[ a = 60/b \quad (3) \] Substituting equation (3) into equation (1), we get: \[ 135 = (60/b) b^3 \] This simplifies to: \[ 135 = 60b^2 \] Now, divide both sides by 60: \[ b^2 = \frac{135}{60} = \frac{27}{12} = \frac{9}{4} \quad (4) \] Taking the square root of both sides gives us: \[ b = \frac{3}{2} \] Now, substitute \( b \) back into equation (3) to find \( a \): \[ a = 60 / \left(\frac{3}{2}\right) = 60 \times \frac{2}{3} = 40 \] Thus, the equation of Ryan's function is: \[ f(x) = 40 \left(\frac{3}{2}\right)^x \]