What is a function's domain?
All of the possible x-values the function could have.
What is a function's range?
All of the possible y-values a function could have.
How can you show that a graph is a function?
You use the vertical line test. That means, you imagine drawing all possible vertical lines on the graph. If any vertical line would intersect the function more than once, it's not a function.
What does this mean?
It can also be written as f(g(x)). This means you put the whole g(x) function inside the brackets of "f". Then you use this as the argument for the "f" function.
What is an inverse function?
It's denoted with a -1 exponent after the function letter. And it basically undoes a function. So if f(3)=5, this means we put in 3 and we get out 5. Therefore, f-1(5)=3. We put in 5 and get out 3. It's exactly the reverse of the original function--it literally swaps the roles of x and y.
How is the domain and range of a function related to the domain and range of the inverse?
- The domain of the original function is the range of the inverse.
- The range of the original function is the domain of the inverse.
How do you find the inverse f-1(x) of the function f(x)?
- Write down f(x), but use the letter "y" rather than "f(x)".
- Swap the "x" and "y" letters.
- Rearrange the equation so the "y" is alone.
- Low key replace the "y" with "f-1(x)" and pretend you never turned it into a y in the first place.
How can you tell if a function has an inverse just by looking at the graph?
You use the horizontal line test. That means, you imagine drawing all possible horizontal lines on the graph. If any horizontal line would intersect the function more than once, there is no inverse.
(That's 'cause when you swap the x and y, this morphs into the vertical line test for functions.)
How do you sketch the inverse of a function?
The inverse is a reflection of the original function over the line y=x. The easiest way to sketch it is to:
- Choose a point on the original function. For example: (1,2).
- Swap the values to get the matching point on the inverse function. For example: (2,1).
- Repeat this process for a couple of points and or any asymptotes.
- Connect the dots, trying to match the general shape of the original function. In the end, it should look like a mirror image of the original function over the diagonal line y=x.
How do you shift f(x)=x2 to the right by h?
You put a "-h" in with the x term.
How do you shift f(x)=x2 to the up by k?
You put a "+k" at the end.
How do you mirror f(x)=x2+k over the x-axis?
You multiply the whole function by -1.
How do you mirror f(x)=x2+k over the y-axis?
You multiply the x-part by -1.
How do you vertically stretch f(x)=x2+k by a?
You multiply the whole function by a.
How do you horizontally stretch f(x)=(x-h)2+k by b?
You multiply the whole x-part by a.
Where is the vertex and line of symmetry for
Vertex: (h, a•k)
Line of symmetry: x=h
If I have a point A(x,y) on my function f, where is the image of that point, A', on the inverse function, f-1?
The inverse literally swaps the x and y values.
So if A is (1, 4), then A' is (4, 1).