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Chapter 0: Problem 34

Find the inverse of each function. $$ y=-2 x+5 $$

Short Answer

Expert verified
The inverse function is \( f^{-1}(x) = -\frac{x}{2} + \frac{5}{2} \).

Step by step solution

01

Write the function in inverse notation

To find the inverse of the function, start by rewriting the function with y in terms of x as given. The function is \( y = -2x + 5 \).
02

Swap x and y

To find the inverse, swap the roles of x and y in the original equation. This gives you: \( x = -2y + 5 \).
03

Solve for y

Next, solve for y in the equation \( x = -2y + 5 \). First, isolate the term with y: \( x - 5 = -2y \). Then divide both sides by -2: \( y = \frac{x-5}{-2} \).
04

Simplify the equation

Simplify the obtained equation: \( y = \frac{x-5}{-2} \) can be rewritten as: \( y = -\frac{x}{2} + \frac{5}{2} \).
05

Write the inverse function

The inverse function is typically written as \( f^{-1}(x) \). Thus, the inverse function is: \( f^{-1}(x) = -\frac{x}{2} + \frac{5}{2} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Function Notation
Functions are essential in mathematics for expressing relationships between variables. A function maps every input (x) to a specific output (y).
We write this relationship as: \[ y = f(x) \]
This is called **function notation**. It helps to easily identify the function's input and output.
Whenever you see something like f(x), know that it represents the value of the function at x.
For example, in the function given in the exercise: \[ y = -2x + 5 \]
The output y depends on the input x. To find the inverse, we need to swap and rearrange these variables.
This is where understanding function notation becomes very useful.
Solving Equations
The process of finding the inverse involves solving equations. This requires familiarity with basic algebraic manipulations.
Follow these steps when solving equations:
  • Identify the variable to solve for (in this case, y).
  • Perform operations to isolate this variable.
  • Simplify the resulting expression.

For instance, after swapping x and y in the function \(y = -2x + 5\), we get:
x = -2y + 5
The goal now is solving for y. We isolate the y-term:
x - 5 = -2y
Then divide by -2:
y = \frac{x - 5}{-2}
This is crucial for correctly identifying the inverse.
Trigonometry
While trigonometry isn't directly part of this specific exercise, the concept of inverse functions extends to trigonometry as well.
Often, trigonometric functions like sine (sin), cosine (cos), and tangent (tan) have inverses: arcsin, arccos, and arctan.
For example, if \( y = \text{sin}(x) \), its inverse is \( x = \text{arcsin}(y) \).
This means if you know the sine of an angle, you can use the arcsine to find the angle itself.
Understanding how to find and use inverse functions is essential across many areas of math, including trigonometry.
By grasping these core principles, you can solve a wide range of problems involving functions and their inverses.

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