Reducing
Simplify the rational expression \( \frac{2x^2 - 5x - 3}{x^2 - 4} \)
Cancelling the Common Factors
Simplify the rational expression \( \frac{2x^2y^3}{4x^3y} \)
Rewriting in Standard Form
Rewrite the rational expression \(\frac{3x^2 + 5x - 2}{x^2 - 4}\) in standard form.
Operations on Rational Expressions
If \( f(x) = \frac{x^2 - 4x + 4}{x - 2} \), simplify \( f(x) \).
Determining if the Point is a Solution
Is the point (2,4) a solution to the equation \(y = \frac{{3x - 2}}{{x + 1}}\)?
Finding the Domain
Find the domain of the function \(f(x) = \frac{1}{x^2 - 4}\).
Solving over the Interval
Solve the given rational equation for \( x \) over the interval \( [0, 2\pi] \): \( \frac{1}{x} = \sin(x) \)
Finding the Range
Given the rational function \( f(x) = \frac{(x-1)}{(x+2)} \), find the range of \( f(x) \).
Finding the Domain and Range
Question 3
Let $f(x)=\frac{3 x-6}{x^{2}+8 x+15}=\frac{(3 x-6)}{(x+5)(x+3)}$
Find:
1) the domain in interval notation
Note: Use -oo for $-\infty$, oo for $\infty$, U for union.
2) the $y$ intercept at the point
3) $x$ intercepts at the point(s)
4) Vertical asymptotes at $x=$
5) Horizontal asymptote at $y=$
Solving Rational Equations
Solve the rational equation: \(\frac{2x}{x - 1} - \frac{3}{x} = 1\)
Adding Rational Expressions
2. $\frac{4 x^{2}-4 x-9}{(2 x+1)(x-1)} \equiv A+\frac{B}{2 x+1}+\frac{C}{x-1}$
a Find the values of the constants $A, B$ and $C$.
b Hence, or otherwise, expand $\frac{4 x^{2}-4 x-9}{(2 x+1)(x-1)}$ in ascending powers of $x$, as far as the $x^{2}$ term
c Explain why the expansion is not valid for $x=\frac{3}{4}$.
Subtracting Rational Expressions
Subtract the rational expressions \(\frac{5x}{x + 2}\) and \(\frac{3x}{x - 2}\)
Multiplying Rational Expressions
Multiply the following rational expressions: \(\frac{4x^{2}-9}{3x^{2}-5x+2}\) and \(\frac{2x-1}{2x^{2}+5x-3}\)
Finding the Equation Given the Roots
Find the equation of a polynomial given the roots are \(2\), \(-3\), and \(1\).
Finding the Asymptotes
Let $f(x)=\frac{3 x-6}{x^{2}+8 x+15}=\frac{(3 x-6)}{(x+5)(x+3)}$
Find:
3) $x$ intercepts at the point(s)
4) Vertical asymptotes at $x=-3$
5) Horizontal asymptote at $y=-5$
Finding the Constant of Variation
If the variation equation is given as \(y = kx^2\), and the point \((2,8)\) lies on the graph of this equation, what is the constant of variation \(k\)?
Finding the Equation of Variation
If y varies directly with x, and y = 8 when x = 2, find the equation of variation.