Power Functions

Definition of Power Functions

A power function refers to a function of the form $y = x^a$ , where $a$ is a constant.

Definition of negative integer powers:

For $x \neq 0$ , positive integer $n$ , we have:

x^{-n} = \frac{1}{x^n}

That is, negative exponent indicates taking the reciprocal.

Examples:

$x^{-1} = \dfrac{1}{x}$
$x^{-2} = \dfrac{1}{x^2}$
$2^{-3} = \dfrac{1}{2^3} = \dfrac{1}{8}$

Properties and Graphs

When $a > 0$ , the function is monotonically increasing on $(0, +\infty)$ .
When $a < 0$ , the function is monotonically decreasing on $(0, +\infty)$ .
When $a$ is even, the function is an even function.
When $a$ is odd, the function is an odd function.

Common Power Functions

Identity Function

The identity function is the most basic power function, corresponding to $a = 1$ . Its graph is a straight line passing through the origin with slope 1, monotonically increasing over the entire real number range, and is an odd function.

Quadratic Power Function

This is the most common quadratic power function, corresponding to $a = 2$ . Its graph is a parabola opening upward, symmetric about the y-axis, and is an even function. It is monotonically increasing when $x > 0$ and $x < 0$ .

Cubic Power Function

The cubic power function corresponds to $a = 3$ . The graph has an inflection point at the origin, is symmetric about the origin, and is an odd function. It is monotonically increasing over the entire real number range.

Inverse Proportion Function

The inverse proportion function corresponds to $a = -1$ . Its graph is a hyperbola, distributed in the first and third quadrants. The domain is $x \neq 0$ , monotonically decreasing when $x > 0$ , and monotonically increasing when $x < 0$ .

$y = x^{-2}$ is not an inverse proportion function. The standard form of inverse proportion function is $y = \dfrac{k}{x}$ (i.e., $y = x^{-1}$ ), while $y = x^{-2}$ is a special case of power function $y = x^a$ (where $a = -2$ ), whose graph and properties are different from the inverse proportion function.

Square Root Function

y = \sqrt{x}

Can be written as $y = x^{1/2}$ , corresponding to $a = 1/2$ . The domain is $x \geq 0$ , and the graph is in the first quadrant, rising slowly as $x$ increases.

Exercises

Exercise 1

Given the power function $y = x^3$ , find its domain, range, and determine its parity.

Answer and Explanation (1 个标签)

power function

Domain: $\mathbb{R}$ ; Range: $\mathbb{R}$ ; $y = x^3$ is an odd function.

Exercise 2

Determine which of the following functions are power functions: $y = 2x^2$ , $y = 3^x$ , $y = x^{-1}$ , $y = \sqrt{x}$ .

Answer and Explanation (1 个标签)

power function

The power functions are $y = 2x^2$ , $y = x^{-1}$ , $y = \sqrt{x}$ (i.e., $y = x^{1/2}$ ). $y = 3^x$ is not a power function.

Exercise 3

Draw the approximate graphs of $y = x^2$ and $y = x^3$ , and compare their monotonicity when $x>0$ and $x<0$ .

Answer and Explanation (2 个标签)

power function graph

$y = x^2$ is monotonically increasing when $x>0$ and $x<0$ , and is an even function; $y = x^3$ is monotonically increasing over the entire real numbers, and is an odd function.

Exercise 4

Find the domain and range of $y = x^{-2}$ .

Answer and Explanation (3 个标签)

power function domain range

Domain: $x \neq 0$ ; Range: $(0, +\infty)$ .

Exercise 5

Determine whether $y = |x|$ is a power function, and explain why.

Answer and Explanation (1 个标签)

power function

No, it is not a power function. $|x|$ cannot be expressed in the form $x^a$ , where $a$ is a constant.

Exercise 6

Given $y = x^{1/3}$ , find the symmetry of its graph about the origin.

Answer and Explanation (2 个标签)

power function parity

$y = x^{1/3}$ is an odd function, and its graph is symmetric about the origin.

Exercise 7

Find the monotonicity of $y = x^a$ (where $a>0$ ) when $x>0$ .

Answer and Explanation (2 个标签)

power function monotonicity

When $x>0$ , $y = x^a$ is monotonically increasing.

Exercise 8

Given $y = x^{-1}$ , find its monotonicity when $x>0$ and $x<0$ .

Answer and Explanation (2 个标签)

power function monotonicity

When $x>0$ , $y = x^{-1}$ is monotonically decreasing; when $x<0$ , $y = x^{-1}$ is monotonically increasing.

Exercise 9

Determine whether $y = x^0$ is a power function, and write its expression.

Answer and Explanation (1 个标签)

power function

$y = x^0 = 1$ (where $x \neq 0$ ), is a constant function, and also belongs to power functions.

Exercise 10

Given $y = x^a$ , if $a$ is even, determine its parity; if $a$ is odd, determine its parity.

Answer and Explanation (2 个标签)

power function parity

When $a$ is even, $y = x^a$ is an even function; when $a$ is odd, $y = x^a$ is an odd function.

Exercise 11

Given the function $f(x) = x^p$ , where $p$ is a constant. If $f(2) = 8$ , $f(4) = 64$ , what is the value of $p$ ?

Answer and Explanation (1 个标签)

power function

From $f(2) = 2^p = 8$ , we get $p = 3$ . Verification: $f(4) = 4^3 = 64$ , which holds.

Exercise 12

Which of the following functions belong to power functions?
(A) $y = x^{1/2}$
(B) $y = 2^x$
(C) $y = x^{-3}$
(D) $y = \ln x$

Answer and Explanation (1 个标签)

power function

(A) and (C) belong to power functions, (B) is an exponential function, (D) is a logarithmic function.

Exercise 13

Given $y = x^a$ , if $a < 0$ , what is the monotonicity of this function on $(0, +\infty)$ ?

Answer and Explanation (2 个标签)

power function monotonicity

It is monotonically decreasing on $(0, +\infty)$ .

Exercise 14

Determine whether $y = x^0$ is a power function, and explain why.

Answer and Explanation (1 个标签)

power function

$y = x^0 = 1$ (where $x \neq 0$ ), is a constant function, and also belongs to power functions.

Exercise 15

Given $y = x^a$ , if $a$ is odd, determine the symmetry of its graph.

Answer and Explanation (2 个标签)

power function parity

It is an odd function, and its graph is symmetric about the origin.

Summary

Symbols Appearing in This Article

Symbol	Type	Pronunciation/Explanation	Meaning in This Article
$x^a$	Mathematical Symbol	x to the power a	General form of power function
$a$	Mathematical Symbol	a	Exponent of power function, a constant
$x^{-n}$	Mathematical Symbol	x to the power negative n	Negative nth power of x, equals $\frac{1}{x^n}$
$\sqrt{x}$	Mathematical Symbol	square root of x	Square root of x
$x^{1/2}$	Mathematical Symbol	x to the power one half	1/2 power of x, equals $\sqrt{x}$
$(0, +\infty)$	Mathematical Symbol	open interval	Open interval from 0 to positive infinity
$\mathbb{R}$	Mathematical Symbol	double-struck R (Real numbers)	Represents the set of real numbers, all real numbers
$y = x^3$	Mathematical Symbol	y equals x cubed	Cubic power function
$y = x^2$	Mathematical Symbol	y equals x squared	Quadratic power function
$y = x^{-1}$	Mathematical Symbol	y equals x to the power negative one	Inverse proportion function

Chinese-English Glossary

Chinese Term	English Term	Phonetic	Explanation
幂函数	power function	/ˈpaʊə ˈfʌŋkʃən/	A function of the form $y = x^a$ , where $a$ is a constant
幂次	exponent	/ɪkˈspəʊnənt/	Indicates how many times a number is multiplied by itself
定义域	domain	/dəʊˈmeɪn/	The set of input values for which the function is defined
值域	range	/reɪndʒ/	The set of output values of the function
奇函数	odd function	/ɒd ˈfʌŋkʃən/	A function satisfying $f(-x) = -f(x)$
偶函数	even function	/ˈiːvən ˈfʌŋkʃən/	A function satisfying $f(-x) = f(x)$
恒等函数	identity function	/aɪˈdentɪti ˈfʌŋkʃən/	The function $f(x) = x$
反比例函数	inverse proportion function	/ˈɪnvɜːs prəˈpɔːʃən ˈfʌŋkʃən/	A function of the form $y = \frac{k}{x}$
平方根函数	square root function	/skweə ruːt ˈfʌŋkʃən/	The function $y = \sqrt{x}$
负指数	negative exponent	/ˈneɡətɪv ˌɛkspəʊˈnɛnt/	An exponent that is a negative number
单调递增	monotonically increasing	/ˌmɒnəˈtɒnɪkəli ɪnˈkriːsɪŋ/	A function where as x increases, y also increases
单调递减	monotonically decreasing	/ˌmɒnəˈtɒnɪkəli dɪˈkriːsɪŋ/	A function where as x increases, y decreases

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