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### Course: Precalculus > Unit 1

Lesson 5: Verifying inverse functions by composition- Verifying inverse functions from tables
- Using specific values to test for inverses
- Verifying inverse functions by composition
- Verifying inverse functions by composition: not inverse
- Verifying inverse functions by composition
- Verify inverse functions
- Composite and inverse functions: FAQ

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# Composite and inverse functions: FAQ

## Why do we need to know how to compose functions?

Composing functions allows us to combine two or more functions into one. We can use this technique to build more complicated functions from simpler ones. It also allows us to model more complex situations in math, physics, engineering, and other disciplines.

## What is an invertible function?

An invertible function is one that can be "undone" or reversed. In other words, we can find another function that will take us back to where we started. For example, a function that adds $2$ to a number is invertible: the inverse function would subtract $2$ from a number.

## Why do we sometimes need to restrict the domain of a function to make it invertible?

In general, we don't always need to restrict the domain of a function to make it invertible. However, there are certain cases where it is necessary.

To be invertible, a function must have the property that each input maps to a unique output. Otherwise, the inverse function won't be able to "undo" the original function unambiguously.

For example, in the function $f(x)={x}^{2}$ , both $-2$ and $2$ map to $4$ . Should the inverse function map $4$ back to $2$ or $-2$ ? It's ambiguous, but a function can only have one output for a given input.

However, if we restrict the domain of $f$ to $x\ge 0$ (non-negative numbers), then the function becomes invertible.

## How do we graph inverse functions?

The easiest way to graph an inverse function is by reflecting the original function across the line $y=x$ . This is because the inputs and outputs switch places when we invert a function.

## Where are composite and inverse functions used in the real world?

These functions are used in a lot of places! For example, in physics, we might use a composite function to model the motion of a falling object. In economics, we might use an inverse function to model the relationship between supply and demand.

## Want to join the conversation?

- Wait so there is a difference between inverse and invertible functions?(19 votes)
- An
**invertible**function is a function that you*can take the inverse of*. Each output of an invertible function is unique. On the other hand, an**inverse**function is a function that undoes the action of another function. Example: f(x)=x+5 is an invertible function because you can find its inverse, which is g(x)=x-5.

Hope this helps!(58 votes)

- in what higher-level fields of math will composite and inverse functions be used?(18 votes)
- composite functions are used everywhere in computer science, engineering, algorithmic analysis (how programs run and how effective they do their job), modeling and simulation of water and air flow, rocket science, weather analysis, etc. It's an essential part of math at the upper level, so having a firm grasp of how to compose and inverse functions is critical for later!(5 votes)

- I'm taking algebra 2 with my juniors and while they do SAT prep me and my soph. friends are teaching radicals to ourselves. What basics do we need to know about radical functions?(5 votes)
- Radical functions involve the use of roots, or radicals, in algebraic expressions. Here are some basic concepts that you should know about radical functions:

1. Simplifying Radicals: You should know how to simplify radical expressions by finding perfect square factors and using the rules of exponents to simplify expressions with radicals.

2. Operations with Radicals: You should know how to add, subtract, multiply, and divide expressions with radicals. When adding and subtracting, you need to have like terms with the same radical, and when multiplying and dividing, you need to use the distributive property and simplify as much as possible.

3. Rationalizing Denominators: You should know how to rationalize denominators, which involves multiplying the numerator and denominator of a fraction by the conjugate of the denominator to get rid of the radical in the denominator.

4. Graphing Radical Functions: You should know how to graph radical functions by finding the domain, range, intercepts, and asymptotes. For square root functions, the domain is all non-negative real numbers, and the range is all non-negative real numbers.

5. Solving Radical Equations: You should know how to solve equations involving radicals by isolating the radical term and squaring both sides to eliminate the radical, then checking for extraneous solutions.

6. Applications: You should know how to apply radical functions to real-world problems, such as calculating distances, areas, and volumes.

By mastering these concepts, you will have a strong foundation in radical functions and be able to solve more complex problems involving radicals.(11 votes)

- what does it mean if something says -f(x)? How would that change the equation?(7 votes)
- I assume -f(x) would reflect a function across some value of x.(6 votes)

- When are other times this type of math would be used.(4 votes)
- when can I use the inverse of functions(2 votes)
- They're helpful in domain and range questions (It's easier to find the domain of your inverse function than the range of your original function)(5 votes)

- Please give me the real world example where we use inverse of a function(1 vote)
- It's basically reverse engineering.(6 votes)

- whats the difference between composite functions and piecewise?(2 votes)
- A composite function uses the output of one function as the input of another function (e.g. f(x) = sin(ln(x)) ). A piecewise function separates the domain of the function into multiple intervals and uses a different function over each interval (e.g. g(x) = sin(x) when x <= 0 and g(x) = ln(x) when x > 0).(3 votes)

- Why can't we have some functions that map to multiple output values for the same input? For example, wouldn't it be reasonable to set the inverse function of x^2 = 4 to x = +- 2?(1 vote)
- By definition a function can only have one output for each input.(1 vote)

- why do you make things so complicated(0 votes)