Questions: Construct an exponential function that has initial value 45 and growth factor 1.75. f(x)=

Transcript text: Construct an exponential function that has initial value 45 and growth factor 1.75. \[ f(x)= \] $\square$

Solution

Solution Steps

To construct an exponential function with a given initial value and growth factor, we use the general form of an exponential function: $ f(x) = a \cdot b^x $, where $ a $ is the initial value and $ b $ is the growth factor. In this case, the initial value $ a $ is 45, and the growth factor $ b $ is 1.75.

Step 1: Define the Exponential Function

The exponential function is defined as: \[ f(x) = a \cdot b^x \] where $ a $ is the initial value and $ b $ is the growth factor. Given $ a = 45 $ and $ b = 1.75 $, we can express the function as: \[ f(x) = 45 \cdot (1.75)^x \]

Step 2: Calculate the Function Value at $ x = 2 $

To find the value of the function at $ x = 2 $, we substitute $ x $ into the function: \[ f(2) = 45 \cdot (1.75)^2 \] Calculating $ (1.75)^2 $: \[ (1.75)^2 = 3.0625 \] Thus, \[ f(2) = 45 \cdot 3.0625 = 137.8125 \]