Intercept of rational function
An intercept of rational function is a point where the graph of the rational function cuts the X or Yaxis. 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 xaxis.
The point where graph intersect the Xaxis for that ycoordinate is zero and the point where the graph intersect the Yaxis for that xcoordinate is zero.
Steps to find the intercept of rational function
To find xintercept :
1) Plug in y=0
2) Find the value of x. Sometimes directly,sometimes by factoring.
To find yintercept :
1) Plug in x = 0
2) Find the value of y.
Examples:
1) Find the xintercept of the $y=\frac{(x2)(x+1)}{(x3)}$ rational function.
Solution: $y=\frac{(x2)(x+1)}{(x3)}$
To find xintercept , plug in y= 0
0 =$y=\frac{(x2)(x+1)}{(x3)}$
So, (x2)(x+1) = 0 ( cross multiply)
x2 =0 and x+1 = 0
x= 2 and x = 1
∴ xintercepts are (2,0) and (1,0)
2) Find the xintercept of $y = y=\frac{(x^{2}5x + 6)}{(x1)}$.
Solution : $y=\frac{(x^{2}5x + 6)}{(x1)}$
To find xintercept , plug in y= 0
0 =$y=\frac{(x^{2}5x + 6)}{(x1)}$
So, $y=x^{2}5x + 6$ = 0 ( cross multiply)
Now the equation is quadratic, so we will find the factors of it.
(x3)(x2)=0
x3 =0 and x2 = 0
x= 3 and x = 2
∴ xintercepts are (3,0) and (2,0)
3) Find the yintercept of $y = y=\frac{(x^{2}5x + 6)}{(x1)}$.
Solution : $y=\frac{(2x^{2}4)}{(x4)}$
To find yintercept , plug in x= 0
0 =$y=\frac{(2x^{2}4)}{(x4)}$
So, $y=\frac{(2(0)^{2}4)}{(04)}$
$y=\frac{4}{4}$
y=1
∴ yintercepts is (0,1)
4) Find the yintercept of the $f(x)=\frac{(x+2)}{(x+5)}$ rational function.
Solution: $f(x)=\frac{(x+2)}{(x+5)}$
To find yintercept , plug in x= 0
$y=\frac{(0+2)}{(0 +5)}$
$y=\frac{2}{5}$
∴ yintercepts are (0,$\frac{2}{5}$)
Practice Questions
Q.1 Find x and y intercepts of the given rational function.
1) $f(x)=\frac{(2x^{2}1)}{(x1)}$
2) $y =\frac{(x^{2}+ x 12)}{(x^{2}9)}$
3) $f(x)=\frac{3}{(x1)}$
11th grade math
Precalculus
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