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Exercises 1.12 Exercise

Sage Plot.

1.

Determine whether the curve is the graph of a function of x.

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2.

Determine whether each of the following function is even, odd or neither even nor odd.

\(\displaystyle f(x) =x\)
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\(\displaystyle f(x) = x^3+x\)
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\(\displaystyle f(x) =\cos x\)
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\(\displaystyle g(x) =1-x^2\)
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\(\displaystyle f(x) =\sin x\)
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\(\displaystyle h(x) = 2x+3\)
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\(\displaystyle f(x) = e^x\)
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\(\displaystyle f(x)=\ln |x| +1\)
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3.

Find the domain of the following functions

\begin{equation*} f(x) =\frac{x-2}{3-x} \end{equation*}

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\begin{equation*} f(x) = \sqrt{4-x^2} \end{equation*}

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\begin{equation*} f(x) =\frac{1}{\sqrt{1-x}} \end{equation*}

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\begin{equation*} f(x) =\sqrt{4-x}+\frac{1}{\sqrt{x^2-1}} \end{equation*}

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\begin{equation*} f(x) = \sqrt[3]{t-1} \end{equation*}

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\begin{equation*} g(x) = \sqrt{5-2x} \end{equation*}

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\begin{equation*} f(x) = \frac{x^4}{x^2+4x-5} \end{equation*}

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4.

Draw graph of the given function \(f(x)=\log_2{x-1}\) and find its domain and Range.

5.

Find the domain of

\begin{equation*} f(x) = \frac{1}{\sqrt{4x-x^2}} \end{equation*}

Answer.

\(4 \lt x \lt 0\text{.}\) Domain: (0,4)

6.

Find the domain and sketch the graph of the following functions:

\begin{equation*} f(x) = \sqrt{2x-3} \end{equation*}

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\begin{equation*} f(x) = \frac{1}{x+1} \end{equation*}

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\begin{equation*} f(x) = \sqrt{-\frac{1}{x}} \end{equation*}

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\begin{equation*} f(x) = \frac{1}{(x^2-1)} \end{equation*}

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\begin{equation*} f(x) = \sqrt[3]{x} \end{equation*}

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\begin{equation*} f(x) = \sqrt[4]{x} \end{equation*}

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\begin{equation*} y=\frac{x^2-1}{x^2+1} \end{equation*}

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Hint.

\begin{equation*} x = \pm \sqrt{-\frac{(1+y)}{(y-1)}} \end{equation*}

Define function,

\begin{equation*} -\frac{(1+y)}{(y-1)} \geq 0 \end{equation*}

\begin{equation*} or, \, \frac{(1+y)}{(y-1)} \leq 0 \end{equation*}

\begin{equation*} \Rightarrow \,y \in [-1,1) \end{equation*}

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7.

Find the domain and sketch the graph of the following functions:

\begin{equation*} f(x) = \sqrt{-x} \end{equation*}

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\begin{equation*} h(x) = |2x-1| \end{equation*}

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\begin{equation*} f(x) = \frac{x}{|x|} \end{equation*}

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\begin{equation*} f(x) = \frac{x^2+x-6}{x+2} \end{equation*}

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\begin{equation*} f(x) = \begin{cases} 1-x^2; & \text{ if } x\leq 2 \\ 3x-5 & \text{ if } x\gt 2 \end{cases} \end{equation*}

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8.

Match each equation with its graph:

\(\displaystyle y=x^6\)
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\(\displaystyle y=\log_6 x\)
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\(\displaystyle y=2+\sin x\)
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9.

Match each equation with its graph:

\(\displaystyle y=x^7\)
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\(\displaystyle y=7^x\)
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\(\displaystyle y=-\frac{1}{x}\)
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\(\displaystyle y=\sqrt[4]{x-1}\)
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10.

Graph each function:

\(\displaystyle y=-\frac{1}{x}\)
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\(\displaystyle y=\tan 2x\)
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\(\displaystyle y=\cos \frac{x}{2}\)
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\(\displaystyle y=\sqrt[3]{x+2}\)
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\(\displaystyle y=2-\cos x\)
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\(\displaystyle y=-2\sin \pi x\)
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\(\displaystyle y=\frac{2}{x-3}\)
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\(\displaystyle y=2+\frac{1}{x+1}\)
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\(\displaystyle y=|\sin x|\)
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\(\displaystyle y=(x-1)^3+1\)
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\(\displaystyle y=2-\sqrt{x+2}\)
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11.

Part A: Multiple Choice (2 points each)
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1. What is the domain of \(f(x) = \sqrt{4 - x}\text{?}\)
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  1. \(\displaystyle x \geq -4\)
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  2. \(\displaystyle x \leq 4\)
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  3. \(\displaystyle x \geq 4\)
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  4. All real numbers
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2. The range of \(g(x) = 3x^2 - 12\) is:
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3. Find the domain of \(h(x) = \frac{5}{x+7}.\)
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4. For \(f(x) = \sqrt[3]{x-5}\text{,}\) the domain and range are:
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5. The function \(k(x) = \frac{1}{(x-2)(x+4)}\) has domain:
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6. What is the range of \(y = -2(x+3)^2 + 5\text{?}\)
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7. The domain of \(f(x) = \ln(7 - 4x)\) is:
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8. For the piecewise function \(f(x) =\begin{cases} x+1 & x \lt -\frac{2}{4} & -2 \leq x \lt \frac{3}{x^2} - 5 & x \geq 3 \end{cases}\) the range is:
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Part B: Free Response (show work, 5 points each)
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1. Find the domain and range of \(f(x) = \sqrt{9 - x^2}\text{.}\) Write in interval notation.
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  Answer.
  
  domain = [-3, 3]
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  range = [0, 3]
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2. Find the domain of \(g(x) = \frac{\sqrt{x+5}}{x^2 - 16}\text{.}\) Write in interval notation.
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  Answer.
  
  Domain: \([-5, -4) \cup (-4, 4) \cup (4, \infty)\)
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3. Find the domain and range of \(h(x) = 2^{x-3} + 1\)
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4. For the function \(y = -3\sqrt{4-x} + 2\text{,}\) state the domain and range in interval notation.
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12.

Match each equation with its graph.

\(\displaystyle y=x^2\)
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\(\displaystyle y=x^5\)
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\(\displaystyle y=x^8\)
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13.

Match each equation with its graph.

\(\displaystyle y=3x\)
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\(\displaystyle y=x^3\)
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\(\displaystyle y=3^x\)
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\(\displaystyle y=\sqrt[3]{x}\)
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14.

Find the range of:

\begin{equation*} y = \frac{x^2+x+1}{x^2-x+1} \end{equation*}

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\begin{equation*} y = \frac{x^2+x+2}{x^2+x+1} \end{equation*}

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\begin{equation*} y = \frac{x^2-5x+6}{x^2-3x+2} \end{equation*}

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Solution.

Hint.

change the equation in quadratic in \(x\text{.}\) Now quadratic equations in \(x\) are defined for \(D \geq 0.\)
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Answer.

\(y\in (-\infty, \frac{1}{3}]\cup [3,\infty)\)
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\(\displaystyle 1\lt y \leq \frac{7}{3}\)
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\(\displaystyle y\in \mathbb R \setminus \{1,2\}\)
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15.

Discuss whether \(y^2=x^3\) is a function or not?

16.

Find the range of the following functions:

\begin{equation*} f(x) = \frac{1}{x^2-2x-8} \end{equation*}

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\begin{equation*} f(x) = \frac{3}{x^3-16x} \end{equation*}

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\begin{equation*} f(x) = \sqrt{2x^2-5x-7} \end{equation*}

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\begin{equation*} f(x) = \sqrt{8+2x-3x^2} \end{equation*}

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17.

Find the functions \(f\circ g, \, g\circ f, \, f\circ f, \, g\circ g\) and their domains.

\begin{equation*} f(x) = x^2, \, g(x) = 1-\sqrt{x} \end{equation*}

18.

Find the \(f\circ g\circ h\) and their domains.

\begin{equation*} f(x) = x-2, \, g(x) = \sqrt{x}, \, h(x) = x-1 \end{equation*}

19.

Plot the function and its inverse:

\begin{equation*} f(x) = \frac{x^2+5x+6}{x+2} \end{equation*}

20.

Draw graph and state the domain and range of

\begin{equation*} y=\sqrt{9-x^2} \end{equation*}

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\begin{equation*} y=\sqrt{x-1} \end{equation*}

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\begin{equation*} y=|2x| \end{equation*}

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\begin{equation*} y=\frac{1}{x-4} \end{equation*}

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\begin{equation*} y=|2x|-1 \end{equation*}

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Answer.

\begin{equation*} x\in [-3,3]; \, y \in [0,3] \end{equation*}

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\begin{equation*} x\in [1,\infty); \, y \in [0,\infty) \end{equation*}

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\begin{equation*} x\in (-\infty,\infty); \, y \in [0,\infty) \end{equation*}

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\begin{equation*} x\in \mathbb R \setminus \{4\}; \, y \in \mathbb R \setminus \{0\} \end{equation*}

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\begin{equation*} x\in \mathbb R; \, y \in [-1, \infty) \end{equation*}

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21.

Draw graph of \(y=\sin x, \, |\sin x|, \, \sin |x|, \, \text { and } \, |y| =\sin x. \)