Let f(x) = 2x - 5 g(x) = 4x - 1 h(x) = x² + x + 2. Evaluate the indicated function without finding an equation for the function. ƒ^-1 (1)

Verified step by step guidance

1. Identify that the notation ƒ¹(1) represents the value of the inverse function of ƒ at 1, meaning we want to find the input x such that ƒ(x) = 1.
2. Write the equation based on the function ƒ(x) = 2x - 5, setting it equal to 1: 2x - 5 = 1.
3. Solve the equation for x by first adding 5 to both sides: 2x = 1 + 5.
4. Then divide both sides by 2 to isolate x: x = \(\frac{1 + 5}{2}\).
5. The value of x found is ƒ¹(1), which is the output of the inverse function at 1.

Key Concepts

Function Notation and Evaluation

Function notation, such as f(x), represents a rule that assigns each input x to an output. Evaluating a function at a specific value means substituting that value into the function's expression and simplifying to find the output.

Composite Functions and Inverse Notation

The notation ƒ¹(1) often represents the inverse function evaluated at 1, not the reciprocal. Understanding inverse functions involves finding an input that maps to a given output under the original function.

Solving Linear Equations

To find ƒ¹(1), you solve the equation f(x) = 1 for x. This requires isolating x by performing algebraic operations, such as addition, subtraction, multiplication, or division, to find the input that produces the output 1.