# Intercept of rational function

An intercept of rational function is a point where the graph of the rational function cuts the X- or Y-axis. The rational function is in the form of fraction.
$y=\frac{f(x)}{g(x)}, g(x)\neq 0$
In the rational function , if the numerator never be zero then the graph never touches the x-axis.
The point where graph intersect the X-axis for that y-coordinate is zero and the point where the graph intersect the Y-axis for that x-coordinate is zero.

## Steps to find the intercept of rational function

To find x-intercept :
1) Plug in y=0
2) Find the value of x. Sometimes directly,sometimes by factoring.
To find y-intercept :
1) Plug in x = 0
2) Find the value of y.
Examples:
1) Find the x-intercept of the $y=\frac{(x-2)(x+1)}{(x-3)}$ rational function.
Solution: $y=\frac{(x-2)(x+1)}{(x-3)}$
To find x-intercept , plug in y= 0
0 =$y=\frac{(x-2)(x+1)}{(x-3)}$
So, (x-2)(x+1) = 0 ( cross multiply)
x-2 =0 and x+1 = 0
x= 2 and x = -1
∴ x-intercepts are (2,0) and (-1,0)

2) Find the x-intercept of $y = y=\frac{(x^{2}-5x + 6)}{(x-1)}$.
Solution : $y=\frac{(x^{2}-5x + 6)}{(x-1)}$
To find x-intercept , plug in y= 0
0 =$y=\frac{(x^{2}-5x + 6)}{(x-1)}$
So, $y=x^{2}-5x + 6$ = 0 ( cross multiply)
Now the equation is quadratic, so we will find the factors of it.
(x-3)(x-2)=0
x-3 =0 and x-2 = 0
x= 3 and x = 2
∴ x-intercepts are (3,0) and (2,0)

3) Find the y-intercept of $y = y=\frac{(x^{2}-5x + 6)}{(x-1)}$.
Solution : $y=\frac{(2x^{2}-4)}{(x-4)}$
To find y-intercept , plug in x= 0
0 =$y=\frac{(2x^{2}-4)}{(x-4)}$
So, $y=\frac{(2(0)^{2}-4)}{(0-4)}$
$y=\frac{-4}{-4}$
y=1
∴ y-intercepts is (0,1)

4) Find the y-intercept of the $f(x)=\frac{(x+2)}{(x+5)}$ rational function.
Solution: $f(x)=\frac{(x+2)}{(x+5)}$
To find y-intercept , plug in x= 0
$y=\frac{(0+2)}{(0 +5)}$
$y=\frac{2}{5}$
∴ y-intercepts are (0,$\frac{2}{5}$)

## Practice Questions

Q.1 Find x and y intercepts of the given rational function.
1) $f(x)=\frac{(2x^{2}-1)}{(x-1)}$

2) $y =\frac{(x^{2}+ x -12)}{(x^{2}-9)}$

3) $f(x)=\frac{3}{(x-1)}$