Exponential Functions

Definition of Exponential Functions

An exponential function refers to a function of the form $y = a^x$ , where $a > 0$ and $a \neq 1$ .

符号说明

Symbol	Type	Pronunciation/Explanation	Meaning in This Document
$a^x$	Mathematical Symbol	a to the power x	General form of exponential function
$a$	Mathematical Symbol	a	Base of exponential function, must be greater than 0 and not equal to 1
$x$	Mathematical Symbol	x	Exponent of exponential function
$\mathbb{R}$	Mathematical Symbol	Blackboard bold R (Real numbers)	Represents the set of real numbers, all real numbers

Why must $a$ be positive and not equal to 1?

Ensure function values are meaningful and real: When $a > 0$ , $a^x$ has clear real values regardless of whether $x$ is positive, negative, or fractional. If $a < 0$ , such as $(-2)^{1/2}$ , the result is imaginary, which is not suitable for elementary function discussion.
Ensure continuity and monotonicity: When $a > 0$ , $y = a^x$ is continuous and monotonic over the entire real number range. When $a < 0$ , the function graph has discontinuities.
Ensure definition of logarithmic functions: The inverse function of exponential functions is logarithmic functions, which have good definitions only when $a > 0$ and $a \neq 1$ .
Special case of $a = 1$ : When $a = 1$ , $y = 1^x = 1$ , which is a constant function without exponential change characteristics.

Therefore, exponential functions $y = a^x$ require $a > 0$ and $a \neq 1$ to ensure mathematical meaning and good properties.

Properties and Graphs

Domain: $\mathbb{R}$
Range: $(0, +\infty)$
When $a > 1$ , the function is monotonically increasing
When $0 < a < 1$ , the function is monotonically decreasing
Graph passes through point $(0, 1)$
Approaches x-axis asymptotically

Exercises

Exercise 1

Given the exponential function $y = 2^x$ , find its domain and range.

Answer and Explanation (3 个标签)

exponential function domain range

Domain: $\mathbb{R}$ ; Range: $(0, +\infty)$ .

Exercise 2

Determine which of the following functions are exponential functions: $y = 3^x$ , $y = x^3$ , $y = 2^{-x}$ , $y = e^x$ .

Answer and Explanation (1 个标签)

exponential function

$y = 3^x$ , $y = 2^{-x}$ , $y = e^x$ are exponential functions, $y = x^3$ is not.

Exercise 3

Draw approximate graphs of $y = 2^x$ and $y = (1/2)^x$ , and compare their monotonicity.

Answer and Explanation (3 个标签)

exponential function graph monotonicity

$y = 2^x$ is monotonically increasing, $y = (1/2)^x$ is monotonically decreasing.

Exercise 4

Given $y = a^x$ , $a > 1$ , determine its asymptote on the x-axis.

Answer and Explanation (2 个标签)

exponential function asymptote

The x-axis ( $y=0$ ) is its asymptote.

Exercise 5

Determine whether $y = (-2)^x$ is an exponential function, and explain why.

Answer and Explanation (2 个标签)

exponential function definition

No. The base $a$ of an exponential function must be greater than 0 and not equal to 1.

Exercise 6

Given $y = a^x$ , $a > 1$ , if $y(0) = 1$ , $y(1) = 3$ , find the value of $a$ .

Answer and Explanation (1 个标签)

exponential function

$y(1) = a^1 = 3$ , so $a = 3$ .

Exercise 7

Which of the following functions are exponential functions?
(A) $y = 2^x$
(B) $y = x^2$
(C) $y = e^x$
(D) $y = 2^{-x}$

Answer and Explanation (1 个标签)

exponential function

(A), (C), (D) are exponential functions, (B) is not.

Exercise 8

Given $y = a^x$ , $0 < a < 1$ , what is the monotonicity of $y$ ?

Answer and Explanation (2 个标签)

exponential function monotonicity

$y$ is monotonically decreasing.

Summary

Symbols Used in This Article

Symbol	Type	Pronunciation/Explanation	Meaning in This Article
$a^x$	Mathematical Symbol	a to the power x	General form of exponential function
$a$	Mathematical Symbol	a	Base of exponential function, must be greater than 0 and not equal to 1
$x$	Mathematical Symbol	x	Exponent of exponential function
$e^x$	Mathematical Symbol	e to the power x	Natural exponential function, with $e$ as base
$e$	Mathematical Symbol	e	Natural constant, approximately equal to 2.718
$\mathbb{R}$	Mathematical Symbol	Blackboard bold R (Real numbers)	Represents the set of real numbers, all real numbers
$(0, +\infty)$	Mathematical Symbol	Open interval	Left-open right-infinite interval

Chinese-English Glossary

Chinese Term	English Term	Pronunciation	Explanation
指数函数	exponential function	/ɪkspəˈnenʃəl ˈfʌŋkʃən/	Functions of the form $y = a^x$ , where $a > 0$ and $a \neq 1$
底数	base	/beɪs/	The $a$ in exponential functions, must be greater than 0 and not equal to 1
指数	exponent	/ɪkˈspəʊnənt/	The $x$ in exponential functions, indicating power
定义域	domain	/dəʊˈmeɪn/	The range of input values for which the function is defined
值域	range	/reɪndʒ/	The range of output values of the function
单调递增	monotonically increasing	/mɒnəˈtɒnɪkli ɪnˈkriːsɪŋ/	Function values increase as the independent variable increases
单调递减	monotonically decreasing	/mɒnəˈtɒnɪkli dɪˈkriːsɪŋ/	Function values decrease as the independent variable increases
渐近线	asymptote	/ˈæsɪmptəʊt/	A line that the function graph approaches infinitely but never intersects

PreviousPower FunctionsDetailed explanation of the definition, properties, graphs, and typical exercises of power functions. NextLogarithmic FunctionsDetailed explanation of the definition, properties, graphs, and typical exercises of logarithmic functions.

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