Chapter 10 Rational Functions And Further Algebra Flashcards
What is a rational function?
A rational function is a function which can be expressed in the form N(x)/D(x), where N(x) and D(x) are polynomials, and D(x) is not the zero polynomial.
What is the first step to sketch the graph of a rational function?
Find the intercepts, that is where the graph cuts the axes.
What should you examine near vertical asymptotes when sketching a graph?
Examine the behaviour of the graph near the vertical asymptotes; these are the lines x = a if D(a) = 0 and N(a) ≠ 0.
What is the fourth step in sketching a rational function graph?
Show what you have found in Steps 1, 2 and 3 on a sketch graph and complete the sketch.
How can inequalities of the form f(x) > 0 (or f(x) < 0) be solved?
They can be solved by sketching the graph of y = f(x) and finding those parts of the graph which are above (or below) the x-axis.
What happens if p is positive in an inequality?
If p is positive: x > y implies nx > ny.
What must you know when multiplying or dividing both sides of an inequality?
You need to know the sign of the number you are multiplying or dividing by.
What can be done for inequalities of the form f(x) < 0 or f(x) > 0?
They can be solved by finding the critical points (the points where the function is either zero or undefined) and testing whether the inequality is true in each region.
What is an alternative approach to solving an inequality involving a rational function?
An alternative approach is to multiply by the square of the denominator, as this is automatically positive.
