Which graph represents \displaystyle f(x) = \frac{1}{x^2 -1} ?

For the function f(x) = \frac{2}{x + 5} which statement is correct?

A. As x\to \infty, f(x) \to \infty

B. As x\to \infty, f(x) \to 0

C. As x\to 5^+, f(x) \to \infty

D. As x\to 5^-, f(x) \to \infty

For the function f(x) = \frac{x +2}{x -5}, which statement is true?

A. Domain: \{x\in \mathbb{R}, x \neq 5\}, Range: \{y\in \mathbb{R}, y \neq 1\}

B. Domain: \{x\in \mathbb{R}, x \neq -2\}, Range: \{y\in \mathbb{R}, y \neq 1\}

C. Domain: \{x\in \mathbb{R}, x \neq 5\}, Range: \{y\in \mathbb{R}, y \neq 0\}

D. Domain: \{x\in \mathbb{R}, x > 5\}, Range: \{y\in \mathbb{R}, y > 0\}

Write a possible equation for the function in the graph.

Write a possible equation for the function in the graph.

Consider the function f(x) = - \frac{4}{x^2 + 2}.

a) Determine the following key features of the functions:

• i) domain and range
• ii) intercepts
• iii) asymptotes
• iv) Intervals where the function is increasing and intervals where it is decreasing.

b) Sketch a graph of the function.

If f(x) is a polynomial function, does \frac{1}{f(x)} always have a horizontal asymptote? If yes, explain why. If no, provide a counterexample.

Solve the equation.

\displaystyle \frac{3x + 5}{x -4} = \frac{1}{2}

Solve the equation.

\displaystyle \frac{20}{x^2 -4x + 7} = x+ 2

Solve the inequality.

\displaystyle \frac{5}{2x + 3} < 4

Solve the inequality.

\displaystyle \frac{x + 1}{x -2} > \frac{x + 7}{x + 1}

a) Determine an equation of the form f(x) = \frac{ax + b}{cx+ d} for the rational function with x-intercept 2, vertical asymptote at x = -1, and horizongtal asymptote at y = -\frac{1}{2}.

b) Is it possible for another function to have the same key features? If not, explain why not. If so, provide an example.

The acceleration due to gravity is inversely proportional to the square of the distance from the centre of Earth. The acceleration due to gravity for a satellite orbiting 7000 km above the centre of Earth is 8.2 m/s^2.

a) Write a formula for this relationship.

b) Sketch a graph of the relation.

c) At what height will the acceleration due to gravity be 6.0 m/s^2?

When a saw is used to cut wood, a certain percent is lost as sawdust, depending on the thickness of the saw—blade. The wood lost is called the kerf. The percent lost, P(t), can be modelled by the function P(t) = \frac{100t}{t+ W}, where t is the thickness of the blade and W is the thickness of the wood, both in millimetres. Consider a saw cutting a 30-mm-thick piece of wood.

a) Sketch a graph of the function, in the context of this situation.

b) State the domain and range.

c) Explain the significance of the horizontal asymptote.

The electric power, P, in watts, delivered by a certain battery is given by the function

\displaystyle P = \frac{100R}{(2 + R)^2} , where R is the resistance, in ohms.

a) Sketch a graph of this function. b) Describe the power output as the resistance increases from 0 Q to 20 Q.

c) Show that the rate of change is0atR = 2. What does this indicate about the power?

Investigate the graphs of functions of the form f(x) = \frac{1}{x^n} where n \in N. Summarize what happens to the asymptotes and slopes as n increases. Consider, also, when n is even or odd.