Function concepts
A slant asymptote and a decreasing table
- Units 1 and 2
- 6 points
- About 18 minutes
You can use a calculator on this question, just like on exam day.
You work with functions given as a table, a graph described in words, or an equation: composition, inverses, zeros, end behavior and asymptotes, and picking the function type that fits a table. You'll use a graphing calculator. On the exam: Question 1 of 4. Part A of the free-response section: 2 questions in 35 minutes, graphing calculator required.
The question and its sources
A graphing calculator is required for this question. Make sure it is in radian mode. Unless otherwise specified, decimal approximations should be accurate to three places after the decimal point; avoid rounding intermediate values. Unless otherwise specified, the domain of a function f is the set of all real numbers x for which f(x) is a real number. Show the work that leads to your answers where the question asks for it.
Values of f
| x | −1 | 0 | 2 | 3 | 5 |
|---|---|---|---|---|---|
| f(x) | 9 | 4 | 1 | −2 | −6 |
Source: Hypothetical function values
The functions f and g
The function f is decreasing and is defined for all real numbers. The table gives values of f(x) at selected values of x.
The function g is given by g(x) = (2x² + x − 5)/(x − 1).
Source: Hypothetical functions
Suggested time: 18 minutes
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Part (a)
2 points(i) The function h is defined by h(x) = (g ∘ f)(x) = g(f(x)). Find the value of h(0) as a decimal approximation, or indicate that it is not defined. Show the work that leads to your answer. (ii) Find the value of f⁻¹(−2), or indicate that it is not defined.
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Part (b)
2 points(i) Find all real zeros of g, as decimal approximations, or indicate that there are none. (ii) Determine the behavior of g as x approaches 1 from the right. Express your answer using the mathematical notation of a limit.
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Part (c)
2 points(i) Rewrite g(x) in the form mx + k + r/(x − 1), where m, k and r are constants. (ii) The graph of g has a slant asymptote. Write an equation for the slant asymptote, and determine whether the graph of g is above or below this asymptote for x > 1. Give a reason for your answer based on your work in part (c)(i).
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