Mathematics for Management -- Supplementary Electronic Materials

Quiz: Functions

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1. Identify the domain \(D\) of the following relation

\[ \begin{array}{c || c c c c} x & -6 & 0 & 2 & 4 \\ \hline g(x) & 14 & 32 & 38 & 44 \end{array} \] \( D = \{ -6, 14, 2, 28\} \) \( D = \{ -6, 0, 2, 4\} \) \( D = \{ (-6,14), (0,32), (2, 38), (4,44) \} \) \( D = \{ 14, 32, 38, 44 \} \)

2. For the function that gives rise to the set of \((x,y)\)-coordinates \(\{ (0,1), (1,-3), (2,-4), (-4,1) \}\) give both the domain \(D\) and range \(R\).

\( D = \{ 1, -3, -4 \}, R = \{ 0, 1, 2, -4 \} \) \( D = \{ 0, 1, 2, 3, 4 \}, R = \{ 1, -3, -4 \} \) \( D = \{ 0, 1, 2, -4 \}, R = \{1, -3, -4 \} \) \( D = \{ 1, -3, -4 \}, R = \{ 0, 1, 2, 3, 4 \} \)

3. What is the domain of a polynomial function?

It depends on the degree of the function being even or odd \( [0, \infty) \) \( (-\infty, \infty) \) It depends on the vertex

4. What is the domain of the function with \(f(x) = \sqrt{x-10}\)?

\( (-\infty, \infty) \) \( [0, \infty) \) \( [10, \infty) \) \( [0, 10] \)

5. What is the domain of the function with \(f(x) = x^2 + 4\)?

\( [4, \infty) \) \( (-\infty, \infty) \) \( (-\infty, \infty) \setminus \{ 0 \} \) none of these

6. Let \(f: \mathbb{R} \to \mathbb{R}\) be defined as \(f(x) = \left( x^4 \right)^{1/3}\). What is \(F(x) = (f \circ f) (x) = f\left( f(x) \right)\)?

\( F(x) = x \) \( F(x) = x^{4/3} \) \( F(x) = \frac{1}{x^{3/4}} \) \( F(x) = x^{16/9} \)

7. Let \(f: \mathbb{R} \to \mathbb{R}\) be defined as \(f(x) = \left( 3 - x^3 \right)^{1/3}\). What is \(F(x) = (f \circ f) (x) = f\left( f(x) \right)\)?

\( F(x) = x^3\) \( F(x) = x \) \( F(x) = x^{1/3} \) \( F(x) = x^3 - 3\)

8. Let \(f: \mathbb{R} \to \mathbb{R}\) be defined as \(f(x) = 4x+3\). What is \(F(-1) = (f \circ f \circ f) (-1)\)?

\( F(-1) = -1 \) \( F(-1) = 0 \) \( F(-1) = 1 \) \( F(-1) = 2 \)

9. Let \(f(x) = \log_2 \left( \frac{x+4}{x+1} \right)\) and \(g(x) = 2x + x^2\). What is \(F(x) = f\left( g(x) \right)\)?

\( F(x) = 2 \log_2 \left( \frac{x+4}{x+1} \right) \) \( F(x) = 2 \log_2 \left( \frac{x+4}{x+2} \right) \) \( F(x) = 2 \log_2 \left( \frac{x+2}{x+1} \right) \) none of these

10. Let \(f(x) = 2x + 7\) and \(g(x) = x^2 + 7\) for all \(x \in \mathbb{R}\). What are the values of \(x\) for which \(f\left( g(x) \right) = 25\)?

\( -1, 1 \) \( -2, 2 \) \( -\sqrt{2}, \sqrt{2} \) none of these

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Solution: 1b; 2c; 3c; 4c; 5a; 6d; 7b; 8a; 9d; 10c.

Here, a, b, c, d indicate the 1st, 2nd, 3rd, and 4th answer choice, respectively, for the numbered questions.