Aim for FivePrecalculus

Function concepts

A piecewise-linear graph and a rational function

  • 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.

Graph of f (described)

The function f is defined on the closed interval −4 ≤ x ≤ 5. Its graph consists of three line segments that connect the points (−4, 3), (−1, −3), (2, 0) and (5, 6), in that order.

The function g is given by g(x) = (x² − 3x − 1)/(x² + 2).

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 ∘ g)(x) = f(g(x)). Find the value of h(3) 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) = −1, or indicate that there are no such values.

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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 end behavior of g as x increases without bound. Express your answer using the mathematical notation of a limit.

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Part (c)

2 points

(i) Determine whether f has an inverse function on its domain. (ii) Justify your answer using the definition of a function and the graph of f described above.

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