Function concepts
A rational function with a hole
- 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 p
| x | 0 | 2 | 4 | 6 | 8 |
|---|---|---|---|---|---|
| p(x) | 3 | 4.8 | 7.68 | 12.288 | 19.6608 |
Source: Hypothetical function values
The functions f and p
The function f is given by f(x) = (2x² − 3x − 9)/(x² − x − 6).
The function p is increasing and is defined for all real numbers. The table gives values of p(x) at selected values of x.
Source: Hypothetical functions
Suggested time: 18 minutes
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Part (a)
2 points(i) The function h is defined by h(x) = (f ∘ p)(x) = f(p(x)). Find the value of h(2) as a decimal approximation, or indicate that it is not defined. Show the work that leads to your answer. (ii) Find all values of x for which f(x) = 3, or indicate that there are no such values.
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Part (b)
2 points(i) The graph of f has a hole. Find the coordinates of the hole. (ii) Determine the behavior of f as x approaches −2 from the right. Express your answer using the mathematical notation of a limit.
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Part (c)
2 points(i) Based on the table, which of the following function types best models p: linear, quadratic, exponential, or logarithmic? (ii) Give a reason for your answer to part (c)(i) based on the relationship between the change in the output values of p and the change in the input values of p. Refer to values in the table in your reasoning.
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