Concept Check. If ƒ(x) = a^x and ƒ(3) = 27, determine each function value. ƒ(2)

Verified step by step guidance

1. Identify the given function: \(f(x) = a^x\), where \(a\) is the base we need to find.
2. Use the given information \(f(3) = 27\) to set up the equation: \(a^3 = 27\).
3. Solve for \(a\) by taking the cube root of both sides: \(a = \sqrt[3]{27}\).
4. Once you find the value of \(a\), substitute it back into the function to find \(f(2)\): \(f(2) = a^2\).
5. Calculate \(a^2\) using the value of \(a\) found in step 3 to determine \(f(2)\).

Key Concepts

Exponential Functions

An exponential function has the form f(x) = a^x, where the base a is a positive constant and the exponent x is a variable. These functions model growth or decay processes and have unique properties such as always being positive and increasing or decreasing depending on the base.

Using Given Function Values to Find the Base

When a function value is given, such as f(3) = 27, you can substitute x = 3 into f(x) = a^x to form an equation a^3 = 27. Solving this equation helps determine the base a, which is essential for evaluating the function at other points.

Evaluating the Function at a Specific Input

Once the base a is known, you can find the function value at any input by substituting that input into f(x) = a^x. For example, to find f(2), calculate a^2 using the previously determined base, which gives the function value at x = 2.