Question

What are vertical and horizontal line tests for 1-1 function?

Answer 1

This type of question depends on the concept of classification of function. Functions can have many classifications depending on the situation and what we want to do with them. 1-1 is one important classification of function. A function is said to be 1-1 if it has exactly one output value for every input value and exactly one input value for every output value. So a function $f\left( x \right)$ is said to be 1-1 if $f\left( {{x}_{1}} \right)=f\left( {{x}_{2}} \right)\Rightarrow {{x}_{1}}={{x}_{2}}$. In other words we can say that, for two equal output values there must be two equal input values. To check whether a given function is 1-1 function, we can use vertical line test and horizontal line test.

Complete step-by-step solution:
Now we have to write the vertical and horizontal line tests for 1-1 function.
We have to take into consideration that a graph of any function always passes the vertical line test. We can say that the vertical line test specifies that any vertical line drawn through the graph of the function passes through that function only once.
Also we have to take into consideration that all functions pass vertical line tests but only 1-1 functions pass horizontal line tests. In the case of horizontal line tests we can say that if any horizontal line drawn through the graph does not cut through the function more than once then the function is said to be 1-1 function.
For example,

Let us consider the graph of the function $y=2x+1$. Also we have drawn horizontal and vertical lines $\text{g and h}$ respectively to the graph. From the graph we can see that the function passes the vertical line test as it cuts the vertical line only once. Also the function $y=2x+1$ passes the horizontal line test as it does not cut the horizontal line g more than once. And hence we can say that the function $y=2x+1$ is 1-1 function.
Hence we can say that a graph will represent a function, if any vertical line drawn crosses it only once and if the function is 1-1 also, then any horizontal line drawn will cross it only once.

Note: In this type of question students have to note that if the graph satisfies vertical line test then it will represent a function and if the graph satisfies horizontal test as well then the function is said to be 1-1. Also students have to take care that if the horizontal line crosses the graph more than once then the function is not 1-1.