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Elementary Functions at a Glance

Elementary functions are the building blocks of most formulas you will see in calculus and its applications, and the five families covered here are the ones you will meet again and again. Power functions $f(x) = x^a$ cover every real exponent, and their domain depends on it: integer powers run over all of $\mathbb{R}$ , while a rational exponent with an even denominator has to be restricted to keep the roots real, which is why $x^{1/2}$ needs $x \ge 0$ . The exponent also fixes the shape — $a > 1$ grows faster, $a = 1$ is a line, $0 < a < 1$ is concave through $(0,0)$ — and a negative exponent turns the function into a reciprocal such as $x^{-1} = \dfrac{1}{x}$ , whose domain excludes $0$ . Exponential functions $f(x) = a^x$ with $a > 0$ , $a \ne 1$ always stay positive, cross $(0,1)$ , and rise or fall according to whether $a$ is above or below $1$ ; their inverses, the logarithmic functions $\log_a x$ , take the domain $(0, +\infty)$ onto all of $\mathbb{R}$ and pass through $(1,0)$ . The trigonometric functions bring periodicity into the picture: $\sin x$ and $\cos x$ are bounded in $[-1,1]$ with period $2\pi$ , while $\tan x$ and $\cot x$ have period $\pi$ and vertical asymptotes, and $\sin$ is odd while $\cos$ is even. Their inverses reverse the process, with $\arcsin x$ taking domain $[-1,1]$ to range $\left[-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right]$ , $\arccos x$ taking domain $[-1,1]$ to range $[0, \pi]$ , and $\arctan x$ taking domain $\mathbb{R}$ to range $\left(-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right)$ .

Exercises

Exercise 1

State the domain and monotonicity of $f(x) = x^{1/2}$ .

Reference Answer

power functiondomainmonotonicity

Approach:

For a square-root power function the radicand must be non-negative, and the monotonicity of $y = x^a$ on the positive half-line is fixed by the sign of the exponent $a$ .

Steps:

For $f(x) = x^{1/2} = \sqrt{x}$ to be defined we need $x \geq 0$ , so the domain is $[0, +\infty)$ .
Since the exponent $a = \frac{1}{2} > 0$ , the function $y = x^a$ is monotonically increasing for $x > 0$ .
Checking the endpoint, $f(0) = 0$ while $f(x) > 0$ for $x > 0$ , so the increase extends to the whole domain.

Answer:

Domain: $[0, +\infty)$ . Increasing on its domain.

Exercise 2

Compare growth: which grows faster as $x \to +\infty$ , $x^3$ or $2^x$ ?

Reference Answer

exponential functionpower functionlimitgrowth rate

Approach:

Compare the two expressions through their ratio; if the ratio tends to $+\infty$ , the exponential term dominates the power term for large $x$ .

Steps:

Form the ratio $\dfrac{2^x}{x^3}$ for $x > 0$ .
Applying l’Hôpital’s rule three times gives $\lim_{x\to+\infty}\dfrac{2^x}{x^3} = \lim_{x\to+\infty}\dfrac{(\ln 2)^3\,2^x}{6} = +\infty$ , because $(\ln 2)^3 > 0$ and $2^x \to +\infty$ .
Since the ratio tends to $+\infty$ , the exponential $2^x$ eventually exceeds $x^3$ by an unbounded factor.

Answer:

Exponential $2^x$ eventually outgrows any power $x^3$ ; $\lim_{x\to+\infty}\dfrac{2^x}{x^3} = +\infty$ .

Exercise 3

Give one even and one odd trigonometric function with their periods.

Reference Answer

trigonometric functioneven functionodd functionperiod

Approach:

Recall that $\sin x$ is odd and $\cos x$ is even, and that both have fundamental period $2\pi$ ; this settles both the parity and the period in one step.

Steps:

For $y = \cos x$ we have $\cos(-x) = \cos x$ , so it is an even function, and its minimum positive period is $2\pi$ .
For $y = \sin x$ we have $\sin(-x) = -\sin x$ , so it is an odd function, and its minimum positive period is $2\pi$ .
Both graphs therefore repeat themselves once every $2\pi$ radians.

Answer:

Even: $\cos x$ , period $2\pi$ . Odd: $\sin x$ , period $2\pi$ .

Summary

Symbols Used in This Article

Symbol	Type	Pronunciation/Explanation	Meaning in This Article
$x^a$	Mathematical symbol	x to the a	The expression of a power function
$a^x$	Mathematical symbol	a to the x	The exponential function
$\log_a x$	Mathematical symbol	log base a of x	The logarithmic function
$\pi$	Greek letter	Pi (pie)	The period constant of the trigonometric functions
$\arcsin x$	Mathematical symbol	arc-sine of x	The inverse trigonometric function
$\arctan x$	Mathematical symbol	arc-tangent of x	The inverse trigonometric function

Chapters

Function Inflection Points

Definition and Classification of Elementary Functions

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Elementary Functions at a Glance

Exercises

Exercise 1

Exercise 2

Exercise 3

Summary

Symbols Used in This Article

Chapters

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