# Point of discontinuity

##### Examples

###### Lessons

**Investigating How a Point of Discontinuity Appears on a Graph**

Sketch and compare the following two functions:

i) $f\left( x \right) = 2x + 5$

ii) $g\left( x \right) = \frac{{2{x^2} + 11x + 15}}{{x + 3}}$**Sketching Rational Functions Incorporating Asymptotes and Points of Discontinuity**

Sketch the rational function: $f\left( x \right) = \frac{{2{x^2} - 7x + 5}}{{2{x^3} - 11{x^2} + 19x - 10}}$

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###### Topic Notes

• "point of discontinuity" exists when the numerator and denominator have a factor in common.

i.e. $\left( x \right) = \frac{{ - \left( {3x - 8} \right)\left( {x + 5} \right)\left( {2x - 7} \right)}}{{\left( {x + 5} \right)\left( {4x + 9} \right)\left( {3x + 8} \right)\left( {2x - 7} \right)}}$ ; points of discontinuity exist at $x = - 5$ and $x = \frac{7}{2}$ .

• To determine the coordinates of the point of discontinuity:

1) Factor both the numerator and denominator.

2) Simplify the rational expression by cancelling the common factors.

3) Substitute the non-permissible values of x into the simplified rational expression to obtain the corresponding values for the y-coordinate.

i.e. $\left( x \right) = \frac{{ - \left( {3x - 8} \right)\left( {x + 5} \right)\left( {2x - 7} \right)}}{{\left( {x + 5} \right)\left( {4x + 9} \right)\left( {3x + 8} \right)\left( {2x - 7} \right)}}$ ; points of discontinuity exist at $x = - 5$ and $x = \frac{7}{2}$ .

• To determine the coordinates of the point of discontinuity:

1) Factor both the numerator and denominator.

2) Simplify the rational expression by cancelling the common factors.

3) Substitute the non-permissible values of x into the simplified rational expression to obtain the corresponding values for the y-coordinate.

###### Basic Concepts

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