A one-to-one function is an injective function. A function $f:A\rightarrow B$ is an injection if $x=y$ whenever $f(x)=f(y)$. Both functions $f(x)=\dfrac{x-3}{x+2}$ and $f(x)=\dfrac{x-3}{3}$ are injective. Let's prove it for the first one

A one to one function is a function that maps no two elements of its domain to a single value in its range. A one-to-one function can be determined by using the horizontal line test. Also, there are various other ways to determine a one-one function.

To determine that whether the function f(x) is a One to One function or not, we have two tests. 1) Horizontal Line testing: If the graph of f(x) passes through a unique value of y every time, then the function is said to be one to one function.

There's an easy way to look at it, then there's a more technical way. (The technical way will really get us off track, so I'm leaving it out for now.) If you can draw a horizontal line so that it hits the graph in more than one spot, then it is NOT one-to-one. one-to-one? Yep! inverse function.

Prove without using graphing calculators that $f: \mathbb R\to \mathbb R,\,f(x)=x+\sin x$ is both one-to-one, onto (bijective) function.

To prove that a function is one-to-one, you can do it in any of two (equivalent) ways: show that if a and b are any numbers with the property that f(a) = f(b), then it must be the case that a = b; or show that if a ≠ b, then f(a) must be different from f(b).

The line y = 1 intersects the graph of f in one point, and the line y = 3 intersects the graph in zero points. Furthermore, we can see from the figure that each horizontal line will intersect the graph in at most one point. Hence f is a one-to-one function.

Sep 27, 2022 · Use the horizontal line test to recognize when a function is one-to-one. Find the inverse of a given function. Draw the graph of an inverse function. Some functions have a given output value that corresponds to two or more input values. For example, on a menu there might be five different items that all cost $7.99.

Given a graph of a function, use the horizontal line test to determine if the graph represents a one-to-one function. Inspect the graph to see if any horizontal line drawn would intersect the curve more than once. If there is any such line, determine that the function is not one-to-one.

Often you can prove that a function f is one-to-one by setting up the equation f(a)=f(b), trying to solve for a, and finding the only solution is b; if you suspect a function is not one-to-one, try finding an explicit counterexample (as I did above, showing that an even function on a strict superset of the origin is not one-to-one).