Calculus/Precalculus/Solutions

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Convert to interval notation[edit]

1.  \{x:-4<x<2\} \,

\mathbf{(-4,2)}

2.  \{x:-\frac{7}{3} \leq x \leq -\frac{1}{3}\}

\mathbf{[-\frac{7}{3},-\frac{1}{3}]}

3.  \{x:-\pi \leq x < \pi\}

\mathbf{[-\pi,\pi)}

4.  \{x:x \leq 17/9\}

\mathbf{(-\infty, \frac{17}{9}]}

5.  \{x:5 \leq x+1 \leq 6\}

4\leq x\leq5
\mathbf{[4, 5]}

6.  \{x:x - 1/4 < 1\} \,

x<1 \frac{1}{4}=\frac{5}{4}
\mathbf{(-\infty, \frac{5}{4})}

7.  \{x:3 > 3x\} \,

1>x
x<1
\mathbf{(-\infty, 1)}

8.  \{x:0 \leq 2x+1 < 3\}

-1\leq 2x\leq 2
-\frac{1}{2}\leq x<1
\mathbf{[-\frac{1}{2}, 1)}

9.  \{x:5<x \mbox{ and } x<6\} \,

This is equivalent to 5<x<6
\mathbf{(5,6)}

10.  \{x:5<x \mbox{ or } x<6\} \,

It helps to draw a picture to determine the set of numbers described:

5 lt x or x lt 6.png

A number in the set can be on either the red or blue line, so the entire number line is included.
\mathbf{(-\infty,\infty)}

State the following intervals using set notation[edit]

11.  [3,4] \,

\mathbf{\{x:3\leq x\leq 4\}}

12.  [3,4) \,

\mathbf{\{x:3\leq x<4\}}

13.  (3,\infty)

\mathbf{\{x:x>3\}}

14.  (-\frac{1}{3}, \frac{1}{3}) \,

\mathbf{\{x:-\frac{1}{3}<x<\frac{1}{3}\}}

15.  (-\pi, \frac{15}{16}) \,

\mathbf{\{x:-\pi<x<\frac{15}{16}\}}

16.  (-\infty,\infty)

\mathbf{\{x:x\in\Re\}}

Which one of the following is a true statement?[edit]

17.  |x+y| = |x| + |y| \,

Let x=-5, y=5. Then
|x+y|=|-5+5|=|0|=0, and
|x|+|y|=|-5|+|5|=5+5=10
Thus, |x+y| \neq |x| + |y|
false

18.  |x+y| \geq |x| + |y|

Using the same example as above, we have |x+y|\ngeq |x| + |y|.
false

19.  |x+y| \leq |x| + |y|

true

Evaluate the following expressions[edit]

20.  8^{1/3} \,

(2^3)^{1/3}=2^1=\mathbf{2}

21.  (-8)^{1/3} \,

(-2^3)^{1/3}=-2^1=\mathbf{-2}

22.  \bigg(\frac{1}{8}\bigg)^{1/3} \,

(\frac{1}{2^3})^{1/3}=(2^{-3})^{1/3}=2^{-1}=\mathbf{\frac{1}{2}}

23.  (8^{2/3}) (8^{3/2}) (8^0) \,

8^{\frac{2}{3}+\frac{3}{2}+0}=8^{\frac{4}{6}+\frac{9}{6}}=8^{\frac{13}{6}}=(2^3)^{\frac{13}{6}}=\mathbf{2^{13/2}}

24.  \bigg( \bigg(\frac{1}{8}\bigg)^{1/3} \bigg)^7

((\frac{1}{2^3})^{1/3})^7=((2^{-3})^{1/3})^7=(2^{-1})^7=2^{-7}=\frac{1}{2^7}=\mathbf{\frac{1}{128}}

25.  \sqrt[3]{\frac{27}{8}}

(\frac{27}{8})^{1/3}=(\frac{3^3}{2^3})^{1/3}=\frac{3^1}{2^1}=\mathbf{\frac{3}{2}}

26.  \frac{4^5 \cdot 4^{-2}}{4^3}

4^{5-2-3}=4^0=\mathbf{1}

27.  \bigg(\sqrt{27}\bigg)^{2/3}

((3^3)^{1/2})^{2/3}=(3^\frac{3}{2})^\frac{2}{3}=3^1=\mathbf{3}

28.  \frac{\sqrt{27}}{\sqrt[3]{9}}

\frac{(3^3)^{1/2}}{(3^2)^{1/3}}=\frac{3^\frac{3}{2}}{3^\frac{2}{3}}=3^{\frac{3}{2}-\frac{2}{3}}=3^{\frac{9}{6}-\frac{4}{6}}=\mathbf{3^{5/6}}

Simplify the following[edit]

29.  x^3 + 3x^3 \,

\mathbf{ 4x^3 }

30.  \frac{x^3 + 3x^3}{x^2}

\mathbf{ 4x }

31.  (x^3+3x^3)^3 \,

\mathbf{ 64x^9 }

32.  \frac{x^{15} + x^3}{x}

\mathbf{ x^{14} + x^2 }

33.  (2x^2)(3x^{-2}) \,

\mathbf{ 6 }

34.  \frac{x^2y^{-3}}{x^3y^2}

\mathbf{ \frac{1}{xy^5} }

35.  \sqrt{x^2y^4}

\mathbf{ xy^2 }

36.  \bigg(\frac{8x^6}{y^4}\bigg)^{1/3}

\mathbf{ \frac{2x^2}{y^{\frac{4}{3}}}}

Functions[edit]

52. Let f(x)=x^2.

a. Compute f(0) and f(2).

f(0)=0, f(2)=4

b. What are the domain and range of f?

The domain is (-\infty,\infty); the range is [0,\infty),

c. Does f have an inverse? If so, find a formula for it.

No, since f isn't one-to-one; for example, f(-1)=f(1)=1.

53. Let f(x)=x+2, g(x)=1/x.

a. Give formulae for
i. f+g

(f+g)(x)=x+2+1/x=(x^2+2x+1)/x.

ii. f-g

(f-g)(x)=x+2-1/x=(x^2+2x-1)/x.

iii. g-f

(g-f)(x)=1/x-x-2=(1-x^2-2x)/x.

iv. f\times g

(f\times g)(x)=(x+2)/x.

v. f/g

(f/g)(x)=x(x+2) provided x\ne0. Note that 0 is not in the domain of f/g, since it's not in the domain of g, and you can't divide by something that doesn't exist!

vi. g/f

(g/f)(x)=1/[x(x+2)]. Although 0 is still not in the domain, we don't need to state it now, since 0 isn't in the domain of the expression 1/[x(x+2)] either.

vii. f\circ g

(f\circ g)(x)=1/x+2=(2x+1)/x.

viii. g\circ f

(g\circ f)(x)=1/(x+2).

b. Compute f(g(2)) and g(f(2)).

f(g(2))=5/2; g(f(2))=1/4.

c. Do f and g have inverses? If so, find formulae for them.

Yes; f^{-1}(x)=x-2 and g^{-1}(x)=1/x. Note that g and its inverse are the same.

54. Does this graph represent a function? Sinx over x.svg

As pictured, by the Vertical Line test, this graph represents a function.

55. Consider the following function

f(x) = \begin{cases} -\frac{1}{9} & \mbox{if } x<-1 \\ 2 & \mbox{if } -1\leq x \leq 0 \\ x + 3 & \mbox{if } x>0. \end{cases}
a. What is the domain?
b. What is the range?
c. Where is f continuous?

56. Consider the following function

f(x) = \begin{cases} x^2 & \mbox{if } x>0 \\ -1 & \mbox{if } x\leq 0. \end{cases}
a. What is the domain?
b. What is the range?
c. Where is f continuous?

57. Consider the following function

f(x) = \frac{\sqrt{2x-3}}{x-10}
a. What is the domain?
b. What is the range?
c. Where is f continuous?

58. Consider the following function

f(x) = \frac{x-7}{x^2-49}
a. What is the domain?
b. What is the range?
c. Where is f continuous?