Rational Expressions and Functions: math practice, Grade 11 – download the PDF

1 Finding a domain ★★★

Find the domain of \(f(x)=\dfrac{x+4}{x^2-25}\).

2 Simplifying a monomial quotient ★★★

Simplify \(\dfrac{12x^3}{18x^2}\) and state any restriction.

3 Factoring a common factor ★★★

Simplify \(\dfrac{5x+15}{x+3}\). What does the graph of this function look like?

4 Multiplying two fractions ★★★

Multiply \(\dfrac{3x}{8}\cdot\dfrac{4}{9x^2}\) and simplify.

5 Same denominator ★★★

Add \(\dfrac{2x+1}{x+4}+\dfrac{x+7}{x+4}\).

6 Reading asymptotes from the form ★★★

The function \(f(x)=\dfrac{1}{x+3}-2\) is a shifted version of \(y=\dfrac{1}{x}\). Give its vertical and horizontal asymptotes.

7 A cyclist at steady speed ★★★

The distance \(d\) a cyclist covers varies directly with the time \(t\). She rides 42 miles in 3 hours. How far does she ride in 5 hours at the same pace?

8 A trinomial over a difference of squares ★★★

Simplify \(\dfrac{x^2-6x+8}{x^2-4}\) and state the restrictions.

9 Dividing rational expressions ★★★

Compute \(\dfrac{x^2-9}{x+2}\div\dfrac{x+3}{x^2-4}\) and state the restrictions.

10 Subtracting with different denominators ★★★

Subtract: \(\dfrac{5}{x-2}-\dfrac{3}{x+4}\).

11 A first rational equation ★★★

Solve \(\dfrac{5}{x}=\dfrac{3}{x-4}\).

12 Sketching a rational function ★★★

Let \(f(x)=\dfrac{3x}{x+1}\).

1. Find the vertical and horizontal asymptotes.
2. Find the intercepts.
3. Compute \(f(2)\) and \(f(-3)\), then describe the graph.

13 Painters on a fence ★★★

The time \(t\) needed to paint a fence varies inversely with the number \(n\) of painters. Eight painters need 9 hours. How long will 12 painters need?

14 True or false? ★★★

Decide whether each statement is true or false, and justify your answer.

1. \(\dfrac{x+5}{x+7}=\dfrac{5}{7}\) for all \(x\neq -7\).
2. \(\dfrac{x^2-1}{x-1}=x+1\) for all \(x\neq 1\).
3. \(\dfrac1x+\dfrac1y=\dfrac{2}{x+y}\).

15 A complex fraction ★★★

Simplify \(\dfrac{\dfrac1x+\dfrac13}{\dfrac1x-\dfrac13}\) and give the restrictions.

16 Spot the extraneous solution ★★★

Solve \(\dfrac{x}{x-4}-2=\dfrac{4}{x-4}\). Check your answer carefully.

17 A rational equation that becomes a quadratic ★★★

Solve \(\dfrac1x+\dfrac{1}{x+3}=\dfrac12\).

18 Two hoses fill a pool ★★★

Hose A fills a pool alone in \(t\) hours. Hose B alone needs 6 hours more. Together they fill it in 4 hours. How long does each hose need alone?

19 Designing a function ★★★

Write a rational function with vertical asymptote \(x=2\), horizontal asymptote \(y=3\) and \(x\)-intercept \(-1\). Then find its \(y\)-intercept.

20 Hole or asymptote? ★★★

Let \(g(x)=\dfrac{x^2-x-6}{x-3}\).

1. Simplify \(g\) and locate the hole.
2. Does \(g\) have a vertical asymptote?
3. Find the \(y\)-intercept and sketch the graph.

21 Combined variation ★★★

The quantity \(y\) varies directly with \(x\) and inversely with \(z\). When \(x=4\) and \(z=10\), \(y=6\). Find \(y\) when \(x=7\) and \(z=5\).