Understanding Inverse Functions
What is an inverse function?
It's a function that "undoes" another function. If f(x)=y, then f-1(y)=x.
How can you check if a function has an inverse using a graph?
Use the horizontal line test: if any horizontal line touches the graph more than once, it doesn’t have an inverse.
What’s the inverse of f(x)=3x+5?
f-1(x)=x-5/3
If you have f(x)=4x−7, how can you solve f(x)=13?
Use the inverse: f-1(x)=x+7/4; so, f-1(13)=5
What does a function and its inverse look like on a graph?
One graph is the mirror image of the other across the line y=x.
How do you know if a function has an inverse?
The function must be one-to-one (each output has only one input).
Why is the line y=x important?
It’s the line of reflection between a function and its inverse.
What’s the inverse of f(x)=x−2/4
f-1(x)=4x+2
You know f-1(x)=x−3/2, what does that tell you about f(x)?
f(x)=2x+3
How does the inverse change the graph?
It flips the input and output (x and y switch places).
How are a function and its inverse related on a graph?
They are mirror images over the line y=x.
Can a function have more than one inverse?
No. A function can only have one inverse.
Does f(x)=2x/x+1 have an inverse? If so, what is it?
Yes. f-1(x)=x/2−x
If f(x)=x2 and you're solving f(x)=16, what’s the inverse step?
Take the square root: x=√16=4(or x=−4, depending on context)
Give a real-world example of an inverse function.
Converting Celsius to Fahrenheit and vice versa.
What does "one-to-one" mean?
It means no two inputs have the same output.
What do the graphs of a function and its inverse look like?
They are symmetrical across the line y=x.
If f(x)=2x+1, what is f-1(5)?
f-1(5)=2
If a machine transforms input x into 5x+2, what input gives output 27?
Solve with inverse: f-1(27)=27−2/5=5
If f(3)=7, what is f-1(7)?
f-1(7)=3
What is the inverse of the function f(x)=xf(x)=x?
It’s the same: f-1(x)=x
If f(g(x))=x, what does that mean?
f and g are inverses of each other.
How can you check if two functions are inverses?
Plug one into the other: if both f(f-1(x))=x and f-1(f(x))=x, they’re inverses.
How are inverse functions useful in solving real-world problems?
They help “work backwards,” like finding the original price before tax or reversing formulas.
Why do we use inverse functions in math and real life?