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### Course: Algebra (all content)>Unit 7

Lesson 22: Determining whether a function is invertible (Algebra 2 level)

# Restricting domains of functions to make them invertible

Sal is given the graph of a trigonometric function, and he discusses ways in which he can change the function to make it invertible. Created by Sal Khan.

## Want to join the conversation?

• Don't you use the vertical line test?
• No. The vertical line test just ensures that the relation is a function, not that it is invertible. To make sure it is one-to-one, you need to use the horizontal line test.
• I don't have a solid background about functions. Not yet at least. Do you think I can get through this lesson or should I do some Algebra and then come back?
• Start with algebra 1 then move on to algebra 2 and once you master both then come to precalculus. I recommend doing the high school geometry and if you can then trigonometry along the way for some variety. This is what I’ve been doing for the past 4 years along with physics, chemistry, and astronomy. If any of these are hard then there is a whole list of “Get ready for _____” courses to help out. So if algebra 1 is hard then either go to pre algebra or get ready for algebra 1 and same goes for high school geometry if it is too hard then do basic geometry or get ready for geometry. Oh and I forgot to mention, if you prefer studying grade by grade then you have that option too by taking the grade courses which teach you everything you will learn in that year.
• how do you know that the set is open or closed line
• A set can be open, closed, or part open or part closed, depending on whether its endpoints are included.
Using PARENTHESES when writing the interval in question indicates "OPEN" (in other words, the endpoint is NOT included).
Using BRACKETS when writing the interval in question indicates "CLOSED" (in other words, the endpoint IS included).
Using parentheses AND brackets when writing the interval in question indicates that only one of the endpoints is included.
• Hello, So can I say, I Must find an interval with one direction/slop (negative or positive, but not both in the same interval) to make possible the reversal .

Forgive my approximate English, I'm French

thanks.
• Exactly. In more precise mathematical language, we can say that we are finding an interval for which our function is monotonic. The function is invertible within that interval.
• What is the difference between [intervals] and (intervals) - intervals with brackets and parentheses.
• [Intervals] with brackets are closed intervals, that is they include the endpoints.
For example [0,1] is the same as 0 ≤ x ≤ 1
(Intervals) with parentheses are open intervals, that is they do not include the endpoints.
For example (0,1) is the same as 0 < x < 1
We can also combine the two ideas (the brackets don't have to match).
So (0,1] is the same as 0 < x ≤ 1
• Can someone please explain to me the concept of the vertical line test?
(1 vote)
• The vertical line test determines whether or not the graph of a relation (set of points in the coordinate plane) is a function.

If every vertical line in the coordinate plane passes through no more than one point on the graph, then the graph is a function (because this would mean that every input would have no more than one output).

On the other hand, if at least one vertical line in the coordinate plane passes through at least two points on the graph, then the graph is not a function (because such a vertical line would show an example of at least two outputs for the same input).

Have a blessed, wonderful day!
• How can I find the intervals without a graph?
• what's a vertical line test?
• If you have a graph, the vertical line test is a way to visually see if a graph is a function or not. If you can draw a vertical line anywhere in the graph and only pass thru one point on the graph, then you have a function. If a vertical line can pass thru more than one point, this means you have different X-values with the same Y-value. So, the graph can't be a function.

Hope this helps.
• By using the restricted domain mentioned in the video, wouldn't some of the range be cut out? Because in class, I learned that the range of the inverse function helps determine which angle the ratio corresponds to. Or does the range values not matter when inverting sine, cosine, and tangent functions?
• If I don't want to use the horizontal line to test, How can I make sure the function is invertible?
For example, f(x)=cos(x-Π/4)
If I meet a question like: [Π/4+0.001,Π/4+Π], [Π/4-0.001,Π/4+Π], make a graph and use horizontal line can't serve the purpose.
(1 vote)
• 𝑓(𝑥) = cos(𝑥 − 𝜋∕4) for 𝑥 ∈ [𝜋∕4 + 0.001, 𝜋∕4 + 𝜋]
is equivalent to
𝑔(𝑥) = cos(𝑥) for 𝑥 ∈ [0.001, 𝜋]

Over this interval, 𝑔(𝑥) is strictly decreasing and therefore invertible.

– – –

Similarly, 𝑓(𝑥) for 𝑥 ∈ [𝜋∕4 − 0.001, 𝜋∕4 + 𝜋]
is equivalent to 𝑔(𝑥) for 𝑥 ∈ [−0.001, 𝜋]

Over this interval, 𝑔(𝑥) starts off as increasing, but then it becomes decreasing, so there are two different values, 𝑎 and 𝑏, such that 𝑔(𝑎) = 𝑔(𝑏), and therefore 𝑔(𝑥) is not invertible.