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Precalculus : Pre-Calculus

Study concepts, example questions & explanations for Precalculus

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12 Diagnostic Tests 380 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #1001 : Pre Calculus

What is the domain of the following function:

$f(x)=\frac{1}{1+\sqrt x}$

Possible Answers:

$\text{The set of real numbers except 0}$

$\text{The set of real numbers}$

$\text{The set of real numbers}$ $\mathbb{R}-\{ 0,1\}$

$\text{The set of real numbers}$ $\{-1,1\}$

$\text{The domain is :}$

$\{ x\in \mathbb{R}}| x \ge 0\}$

Correct answer:

$\text{The domain is :}$

$\{ x\in \mathbb{R}}| x \ge 0\}$

Explanation:

Note that in the denominator, we need to have $x\ge 0$ to make the square root of x defined. In this case $1+\sqrt x$ is never zero. Hence we have no issue when dividing by this number. Therefore the domain is the set of real numbers that are $\ge 0$

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Example Question #45 : Functions

Find the domain of the following function f(x) given below:

$f(x)=\sqrt{ x^3-1}$

Possible Answers:

$(-\infty,-1)\cup (1,\infty)$

$\{x \in \mathbb{R}: x\ge 1\}$

$\mathbb{R}$

$(-\infty, -1)$

$(-\infty, 1)$

Correct answer:

$\{x \in \mathbb{R}: x\ge 1\}$

Explanation:

x^3-1=(x-1)(x^2-x+1) . Since $(x^2-x+1)\ >0$ for all real numbers. To make the square root positive we need to have $x-1\ge 0$ .

Therefore the domain is :

$\{x\in\mathbb{R}: x\ge 1\}=[1,\infty)$

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Example Question #46 : Functions

What is the range of :

f(x)=1+2cos(x)

Possible Answers:

[-1,1]

[0,3]

$\text{The set of all the reals}$

[-1,3]

[-3,3]

Correct answer:

[-1,3]

Explanation:

We know that $-1\le cos(x)\le 1$ . So $-2\le 2cos(x)\le 2$ .

Therefore:

$1-2\le 1+2cos(x)\le 1+2$ .

This gives:

$-1\le 1+2cos(x)\le 3$ .

Therefore the range is:

[-1,3]

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Example Question #46 : Functions

Find the range of f(x) given below:

f(x)=x^2+2x+3

Possible Answers:

${}(-\infty,\infty)$

$(-\infty, 2)$

$[0,\infty)$

$[2,\infty)$

$(0,\infty)$

Correct answer:

$[2,\infty)$

Explanation:

Note that: we can write f(x) as :

x^2+2x+3=(x+1)^2+2 .

Since,

$(x+1)^2\ge 0 $ for all reals.$

Therefore,

$(x+1)^2+2\ge 2.$

So the range is $[2,\infty)$

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Example Question #34 : Relations And Functions

What is the range of :

f(x)=7+cos(3x)

Possible Answers:

[-7,7]

[0,1]

[-1,1]

[6,8]

[0,7]

Correct answer:

[6,8]

Explanation:

We have $-1\le cos(3x)\le 1$ .

Adding 7 to both sides we have:

$6\le cos(3x)\le 8$ .

Therefore $6\le f(x)\le 8$ .

This means that the range of f is [6,8]

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Example Question #35 : Relations And Functions

Find the domain of the following function:

$f(x)=\sqrt{121-x}$

Possible Answers:

[-121,121]

[0,121]

${}(-\infty,\infty)$

$(-\infty,121]$

$[121,\infty)$

Correct answer:

$(-\infty,121]$

Explanation:

The part inside the square root must be positive. This means that we must be $\ge 0$ . Thus

$121-x\ge0$ . Adding -121 to both sides gives $-x\ge -121$ . Finally multiplying both sides by (-1) give:

$x\le 121$ with x reals. This gives the answer.

Note: When we divide by a negative we need to flip our sign.

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Example Question #1001 : Pre Calculus

What is the domain of the function given by:

$f(x)=\sqrt{1+cos^2(x)}$

Possible Answers:

$[-\frac{\pi}{2},\frac{\pi}{2}]$

[-1,1]

$[0,\infty)$

$\text{The set of all real numbers.}$

[0,1]

Correct answer:

$\text{The set of all real numbers.}$

Explanation:

cos(x) is definded for all reals. cos(x) is always between -1 and 1. Thus $1+cos^2(x)\ge0$ . The value inside the square is always positive. Therefore the domain is the set of all real numbers.

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Example Question #1002 : Pre Calculus

Find the domain of

$\frac{x^{23}}{x^{90}+31}$

Possible Answers:

$(-\infty,\sqrt[90]{31})$

$(\sqrt[90]{31},\infty)$

$\text{The set of all real numbers}$

$(-\infty,\sqrt{31})$

$(-\sqrt[90]{31},\sqrt[90]{31})$

Correct answer:

$\text{The set of all real numbers}$

Explanation:

Since $x^{90}+31 \neq 0$

for all real numbers, the denominator is never 0 .Therefore the domain is the set

of all real numbers.

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Example Question #41 : Relations And Functions

Information about Nernst Equation:

http://physiologyweb.com/calculators/nernst_potential_calculator.html

The Nernst equation is very important in physiology, useful for measuring an ion's potential across cellular membranes. Suppose we are finding an ion's potential of a potassium ion at body temperature. Then the equation becomes:

$y = f(x) = 61.5 \log_{10} x$

Where is the ion's electrical potential in miniVolts and is a ratio of concentration.

What is the domain and range of f(x) ?

Possible Answers:

$\text{Domain}=(0, \infty)$

$\text{Range}= (-\infty, \infty)$

$\text{Domain}=(-\infty , \infty )$

$\text{Range}=(-\infty,\infty)$

$\text{Domain}=(0,\infty)$

$\text{Range}= (61.5, \infty )$

$\text{Domain}=(0,\infty)$

$\text{Range}= (0, \infty)$

$\text{Domain}=(1, \infty)$

$\text{Range}=(0,\infty)$

Correct answer:

$\text{Domain}=(0, \infty)$

$\text{Range}= (-\infty, \infty)$

Explanation:

Apart from the multiplication by 61.5 , this function is very similar to the function $f(x)=\log_{10}x$

The logarithmic function has domain x>0, meaning for every value x>0, the function $\log{ x}$ has an output (and for x = 0 or below, there are no values for log x)

The range indicates all the values that can be outputs of the f(x) . When you draw the graph of y = log(x), you can see that the function extends from -infinity (near x = 0), and then extends out infinitely in the positive x direction.

Therefore the Domain: $(0,\infty)$

and the is Range: $(-\infty, \infty)$

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Example Question #1003 : Pre Calculus

Find the domain and range of the given function.

y=x^2

Possible Answers:

Domain: All real numbers

Range: $y\ge0$

Domain: All real numbers

Range: All real numbers

Domain: x>0

Range: All real numbers

Domain: x>0

Range: y>0

Correct answer:

Domain: All real numbers

Range: $y\ge0$

Explanation:

The domain is the set of x-values for which the function is defined.

The range is the set of y-values for which the function is defined.

Because the values for x can be any number in the reals,

and the values for y are never negative,

Domain: All real numbers

Range: $y\ge0$

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Precalculus : Pre-Calculus

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #1001 : Pre Calculus

Example Question #45 : Functions

Example Question #46 : Functions

Example Question #46 : Functions

Example Question #34 : Relations And Functions

Example Question #35 : Relations And Functions

Example Question #1001 : Pre Calculus

Example Question #1002 : Pre Calculus

Example Question #41 : Relations And Functions

Example Question #1003 : Pre Calculus

All Precalculus Resources

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