In Exercises 59-64, 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. ƒ¹ (1)

Verified step by step guidance

1. Step 1: Understand the problem. The notation ƒ⁻¹(1) refers to the inverse function of f(x). This means we need to find the value of x such that f(x) = 1.
2. Step 2: Start with the given function f(x) = 2x - 5. Set f(x) equal to 1, as we are solving for the input x that produces an output of 1.
3. Step 3: Solve the equation 2x - 5 = 1. Add 5 to both sides to isolate the term with x.
4. Step 4: Divide both sides of the resulting equation by 2 to solve for x. This value of x is the solution to ƒ⁻¹(1).
5. Step 5: Verify your solution by substituting the value of x back into the original function f(x) to ensure that it produces an output of 1.

Key Concepts

Function Evaluation

Function evaluation involves substituting a specific input value into a function to determine its output. In this case, we need to evaluate the inverse function ƒ¹(1), which means finding the value of x such that f(x) equals 1. Understanding how to manipulate and interpret function notation is crucial for this process.

Inverse Functions

An inverse function reverses the effect of the original function. For a function f(x), its inverse f⁻¹(x) satisfies the condition f(f⁻¹(x)) = x. To find ƒ¹(1), we need to determine which input x in the function f(x) = 2x - 5 results in an output of 1, thereby illustrating the relationship between a function and its inverse.

Solving Linear Equations

Solving linear equations involves finding the value of the variable that makes the equation true. In this context, we will set the equation f(x) = 1, which translates to 2x - 5 = 1. By isolating x, we can determine the specific input that corresponds to the output of 1, which is essential for evaluating the inverse function.