Question

Given the function \(f(x)=4 x-2\) determine each of the following. a) \(f^{-1}(4)\) b) \(f^{-1}(-2)\) c) \(f^{-1}(8)\) d) \(f^{-1}(0)\)

Short answer

a) \( \frac{3}{2} \) b) 0 c) \( \frac{5}{2} \) d) \( \frac{1}{2} \)

Step-by-step solution

- Understanding Inverse Functions

To find the inverse function, we need to solve the equation that defines the function for the variable x in terms of y. Given the function is \( f(x) = 4x - 2 \), we write y = 4x - 2 and then solve for x in terms of y.

- Isolate x

Starting from \( y = 4x - 2 \):\[ y + 2 = 4x \]\[ x = \frac{y + 2}{4} \]Thus, the inverse function is \( f^{-1}(y) = \frac{y + 2}{4} \).

- Finding \( f^{-1}(4) \)

To find \( f^{-1}(4) \), substitute 4 for y in the inverse function: \[ f^{-1}(4) = \frac{4 + 2}{4} = \frac{6}{4} = \frac{3}{2} \].

- Finding \( f^{-1}(-2) \)

To find \( f^{-1}(-2) \), substitute -2 for y in the inverse function: \[ f^{-1}(-2) = \frac{-2 + 2}{4} = \frac{0}{4} = 0 \].

- Finding \( f^{-1}(8) \)

To find \( f^{-1}(8) \), substitute 8 for y in the inverse function: \[ f^{-1}(8) = \frac{8 + 2}{4} = \frac{10}{4} = \frac{5}{2} \].

- Finding \( f^{-1}(0) \)

To find \( f^{-1}(0) \), substitute 0 for y in the inverse function: \[ f^{-1}(0) = \frac{0 + 2}{4} = \frac{2}{4} = \frac{1}{2} \].

Key concepts

function notation

Function notation is a way to write equations that define a function. For example, if we have a function like f(x) = 4x - 2, we can express this equation using function notation, where f is the function name, and x is the input variable. Function notation helps us understand how a function processes an input to produce an output.
To find the inverse function, we need to reverse this process. Here we swap the roles of x and y. For instance, in the equation y = 4x - 2, we solve for x to find the inverse function, which would allow us to find x when given y. The inverse function is denoted as f^{-1}(x).
Function notation makes it clear what input values we are using and what output values we expect. It is a concise way to communicate how inputs relate to outputs in a function.

solving equations

Solving equations involves finding the value of the variable that makes the equation true. In the context of inverse functions, we start with our function and solve for the variable x in terms of the output y. Let’s consider the function given in the problem, f(x) = 4x - 2.
To solve for the inverse, we write y = 4x - 2 and isolate x:

• Add 2 to both sides: y + 2 = 4x
• Divide both sides by 4: x = (y + 2)/4

This gives us the inverse function f^{-1}(y) = (y + 2)/4. This function tells us the value of x that would produce a particular y when passed through the original function.
For example, to find f^{-1}(4), substitute 4 for y in the inverse function: f^{-1}(4) = (4 + 2)/4 = 3/2. Similarly, this approach is used to find other specific values of the inverse function.

precalculus

Precalculus is a course that prepares students for calculus by covering concepts such as functions, complex numbers, and trigonometry. One vital aspect of precalculus is understanding functions and their inverses.
Inverse functions essentially reverse the roles of the input and output variables. For example, if a function f(x) transforms x into y, the inverse function f^{-1}(x) transforms y back into x.
Understanding inverse functions is crucial as it helps in solving real-world problems where we need to find the input that results in a specific output. This concept is foundational in calculus and other advanced mathematics courses. Having a strong grasp on function notation and solving equations helps in tackling more complex problems efficiently in calculus and beyond.