Power Function

Definition

1.A power function is a function of the form:

$$f(x) = x^a \qquad a \in \mathbb{R}$$

2.The exponent $a$ is a fixed real number and the variable is the base. In an exponential function $g(x) = b^x,$ the base $b > 0$ is fixed and the variable $x$ is the exponent. The condition $b \neq 1$ excludes the constant case. For an arbitrary real exponent the power is defined through the natural logarithm and the exponential function:

$$x^a := e^{a\ln x} \qquad x > 0$$

3.Every power function is defined on the half-line $(0, +\infty),$ regardless of the real exponent. Whether the definition extends to $x \leq 0$ depends on the value of $a.$

4.A constant factor gives the more general form $f(x) = kx^a$ with $k \neq 0.$ Its absolute value $|k|$ rescales the vertical coordinates, and a negative value of $k$ also reflects the graph across the $x$-axis. The domain remains unchanged.

Domain

5.The domain depends on the type of exponent. A rational exponent must first be written in lowest terms as $p/q,$ because the reduced denominator determines whether negative bases are admissible.

7.The reduction to lowest terms cannot be skipped. The exponents $2/6$ and $1/3$ denote the same rational number, and only the reduced form gives the power at a negative base. Reading the reduced exponent as a cube root gives $(-8)^{1/3} = \sqrt[3]{-8} = -2,$ while reading the unreduced one as $\sqrt[6]{(-8)^2}$ would give $2.$ The two readings agree for $x \geq 0,$ and the definition of a real power at a negative base uses the reduced fraction. The same caution applies to the radicals treated in irrational functions.

8.The exponent $a = 0$ is a separate case. The identity $x^0 = 1$ holds for every $x \neq 0,$ so the power function reduces to the constant $1$ on $\mathbb{R} \setminus \{0\},$ and the value at the origin depends on the convention adopted for $0^0.$

Even natural exponents

9.Let $f(x) = x^n$ with $n$ an even positive integer. The domain is $\mathbb{R}$ and the range is $[0, +\infty),$ since an even power of a real number is never negative.

11.The limits at the two ends of the domain coincide:

$$\lim_{x \to -\infty} x^n = \lim_{x \to +\infty} x^n = +\infty$$

12.Every graph of this family passes through $(-1, 1),$ $(0, 0)$ and $(1, 1).$ If $m > n$ are even positive integers, then $x^m < x^n$ for $0 < |x| < 1,$ while $x^m > x^n$ for $|x| > 1.$ The case $n = 2$ is the parabola $y = x^2.$

13.An even power is not injective on $\mathbb{R},$ because $f(-x) = f(x).$ Restricted to $[0, +\infty)$ it becomes a bijection onto $[0, +\infty),$ and its inverse there is the root $y = \sqrt[n]{x}.$

Odd natural exponents

14.Let $f(x) = x^n$ with $n$ an odd integer greater than one. The exponent $n = 1$ gives the identity function $y = x,$ whose graph is the bisector of the first and third quadrants.

16.The limits at the two ends of the domain have opposite signs:

$$\lim_{x \to -\infty} x^n = -\infty \qquad \lim_{x \to +\infty} x^n = +\infty$$

17.Since $n - 1$ is even and positive, the derivative $nx^{n-1}$ is positive on both sides of the origin and vanishes at the origin. Thus the origin is a stationary point but not an extremum. Since the function is a bijection from $\mathbb{R}$ onto $\mathbb{R},$ its inverse $y = \sqrt[n]{x}$ is defined on the whole line, unlike the even case.

Negative integer exponents

18.Let $f(x) = x^{-n} = 1/x^n$ with $n$ a positive integer. Since the denominator vanishes at the origin, the domain is $\mathbb{R} \setminus \{0\}.$ The function has no zeros because its numerator is $1.$

19.The function $x^{-n}$ has the same parity as $x^n,$ because $(-x)^{-n} = (-1)^n x^{-n}.$ When $n$ is even the function is even, its range is $(0, +\infty),$ and the two branches both lie above the $x$-axis. The function increases on $(-\infty, 0)$ and decreases on $(0, +\infty),$ and both branches are convex. When $n$ is odd the function is odd, its range is $\mathbb{R} \setminus \{0\},$ and the branches occupy opposite quadrants. The function decreases on each branch, and it is concave on $(-\infty, 0)$ and convex on $(0, +\infty).$

20.

A function that decreases on each branch need not be decreasing on its disconnected domain. The function $f(x) = 1/x$ decreases on $(-\infty, 0)$ and on $(0, +\infty),$ yet $-1 < 1$ while $f(-1) = -1 < 1 = f(1).$ A decreasing function must reverse every inequality between points of its domain, but $1/x$ does not reverse this inequality for points on different branches.

21.The case $n = 1$ gives $y = 1/x,$ the rectangular hyperbola of equation $xy = 1.$ It is the basic model of a simple pole in the study of rational functions.

22.The limits that determine the asymptotes are:

$$ \begin{align} \lim_{x \to 0^{+}} x^{-n} &= +\infty \\[6pt] \lim_{x \to \pm\infty} x^{-n} &= 0 \end{align} $$

23.The line $x = 0$ is a vertical asymptote and the line $y = 0$ is a horizontal asymptote. The one-sided limit at the origin from the left equals $+\infty$ when $n$ is even and $-\infty$ when $n$ is odd.

Rational and real exponents

24.On the half-line $(0, +\infty)$ the power $x^a$ is defined for every real exponent. All the graphs pass through $(1, 1),$ since $1^a = 1$ for every $a.$

25.The sign of the exponent controls the monotonicity, and the limits at the two ends of the half-line follow from $x^a = e^{a\ln x}$:

$$ \lim_{x \to 0^{+}} x^{a} = \begin{cases} 0 & \text{if } a > 0 \\[6pt] +\infty & \text{if } a < 0 \end{cases} \qquad \lim_{x \to +\infty} x^{a} = \begin{cases} +\infty & \text{if } a > 0 \\[6pt] 0 & \text{if } a < 0 \end{cases} $$

26.For $a > 0$ the function increases from $0$ to $+\infty$ and extends continuously to $x = 0$ by setting $0^a = 0.$ For $a < 0$ it decreases from $+\infty$ to $0.$ For $a = 0$ it is constant.

27.Two exponents can be compared at a fixed base. If $a < b,$ then

$$x^{a} > x^{b} \quad \text{for } 0 < x < 1, \qquad x^{a} < x^{b} \quad \text{for } x > 1$$

28.The order reverses as $x$ passes through $1$ because $x^{b}/x^{a} = x^{b-a}$ with $b - a > 0,$ and a positive power of $x$ is smaller than $1$ when $0 < x < 1$ and larger than $1$ when $x > 1.$

29.For $0 < a < 1,$ the graph is concave, while for $a > 1$ it is convex. The square root $y = x^{1/2}$ belongs to the first group and $y = x^{2}$ to the second. The two curves are reflections of each other across the bisector $y = x,$ since they are mutually inverse on $[0, +\infty).$

Symmetry

30.For an integer exponent $n,$ we have

$$(-x)^{n} = (-1)^{n}x^{n}$$

31.The factor $(-1)^n$ equals $1$ for even $n$ and $-1$ for odd $n,$ so $x^n$ is an even function exactly when $n$ is even and an odd function exactly when $n$ is odd.

32.A rational exponent $p/q$ in lowest terms with $q$ odd admits negative bases, and the same computation applies with $(-1)^{p/q} = (-1)^{p}.$ The function is even when $p$ is even and odd when $p$ is odd. When $q$ is even, or when the exponent is irrational, the domain contains no negative number and the question of symmetry does not arise.

Derivative

33.On $(0, +\infty)$ the derivative formula holds for every real exponent. Writing $x^a = e^{a\ln x}$ and applying the chain rule gives

$$ \begin{align} \frac{d}{dx}x^{a} &= e^{a\ln x}\frac{a}{x} \\[6pt] &= x^{a}\frac{a}{x} \\[6pt] &= ax^{a-1} \end{align} $$

34.For positive integer exponents the formula holds on $\mathbb{R},$ while for negative integer exponents it holds on $\mathbb{R} \setminus \{0\}.$ For $a = 0,$ the derivative is zero wherever the function is defined. For $a \neq 0,$ the derivative is a constant multiple of the power function with exponent $a - 1.$

35.The second derivative determines the concavity on the half-line:

$$\frac{d^{2}}{dx^{2}}x^{a} = a(a-1)x^{a-2}$$

36.Since $x^{a-2} > 0$ for $x > 0,$ the sign of the second derivative is the sign of $a(a-1).$ The product is positive for $a < 0$ and for $a > 1,$ where the graph is convex, and negative for $0 < a < 1,$ where the graph is concave. The exponents $a = 0$ and $a = 1$ give straight lines, whose second derivative vanishes.

37.The first derivative also describes the graph near the origin when $a > 0.$ As $x \to 0^{+},$ the quantity $ax^{a-1}$ tends to $0$ for $a > 1$ and to $+\infty$ for $0 < a < 1.$ The graph therefore meets the origin with a horizontal tangent in the first case and with a vertical half-tangent in the second. In the second case the right difference quotient diverges, so the continuous extension has a point of non-differentiability at the origin.

Integral

38.For $a \neq -1,$ differentiating $x^{a+1}/(a+1)$ gives $x^a.$ Hence the indefinite integral on $(0, +\infty)$ is:

$$\int x^{a} \ dx = \frac{x^{a+1}}{a+1} + c \qquad a \neq -1$$

39.When $a = -1,$ the denominator in the preceding expression vanishes. The primitive in that case is the logarithm:

$$\int \frac{1}{x} \ dx = \ln|x| + c$$

Comparison of growth rates

40.Two power functions with positive exponents both diverge at infinity, and the function with the larger exponent grows faster. For $0 < a < b$:

$$\lim_{x \to +\infty}\frac{x^{a}}{x^{b}} = \lim_{x \to +\infty}x^{a-b} = 0$$

41.The quotient is a power with negative exponent, so it tends to zero, which means $x^a$ is negligible with respect to $x^b.$ Positive powers can also be compared with logarithmic and exponential functions. For every $a > 0$ and every $c > 1$:

$$\ln x = o(x^{a}) \qquad x^{a} = o(c^{x}) \qquad x \to +\infty$$

42.A logarithm grows more slowly than every positive power, and every positive power grows more slowly than every exponential with base greater than one.

Power function and exponential function

43.Power and exponential functions have different scaling laws. For a power function on $(0, +\infty),$ multiplying the input by a factor $\lambda > 0$ multiplies the output by a factor that depends on $\lambda$ alone:

$$f(\lambda x) = (\lambda x)^{a} = \lambda^{a}f(x)$$

44.For an exponential function $g(x) = c^x$ with $c > 0$ and $c \neq 1,$ adding $h$ to the input multiplies the output by $c^h$:

$$g(x + h) = c^{x+h} = c^{h}g(x)$$

45.A power function turns a fixed ratio of inputs into a fixed ratio of outputs, while an exponential function turns a fixed difference of inputs into a fixed ratio of outputs. The area of a square as a function of the length of its side is a power law, while a population that multiplies by a fixed factor over equal intervals obeys an exponential law.

Example

46.We study the function

$$f(x) = x^{5/3} = \sqrt[3]{x^{5}}$$

47.The exponent $5/3$ is in lowest terms with odd denominator and positive numerator, so the domain is $\mathbb{R}.$ The numerator is odd, hence $f$ is an odd function and its graph is symmetric about the origin. The only zero is $x = 0.$

48.For $x \neq 0,$ we have $f(x) = x\sqrt[3]{x^2},$ and the product rule and chain rule give

$$ \begin{align} f'(x) &= \sqrt[3]{x^2} + \frac{2x^2}{3(x^2)^{2/3}} \\[6pt] &= \frac{5}{3}x^{2/3} \\[6pt] &= \frac{5}{3}\sqrt[3]{x^2} \end{align} $$

49.The cube root of a square is positive for every $x \neq 0,$ so $f'$ is positive on both half-lines. At the origin the difference quotient is

$$f'(0) = \lim_{h \to 0}\frac{h^{5/3}}{h} = \lim_{h \to 0}h^{2/3} = 0$$

50.The graph therefore has a horizontal tangent at the origin. The function is strictly increasing on each half-line, and $f(x) < 0$ for $x < 0,$ while $f(0) = 0$ and $f(x) > 0$ for $x > 0.$ These facts show that $f$ is strictly increasing over $\mathbb{R}.$ Since the derivative does not change sign, the origin is a stationary point but not an extremum.

51.For $x \neq 0,$ the second derivative is

$$f''(x) = \frac{10}{9}x^{-1/3} = \frac{10}{9\sqrt[3]{x}}$$

52.It is negative for $x < 0$ and positive for $x > 0,$ so the graph is concave on $(-\infty, 0)$ and convex on $(0, +\infty).$ The second derivative is undefined at $x = 0,$ but its sign changes there, so the origin is an inflection point with horizontal tangent.

53.The limits at the ends of the domain are

$$\lim_{x \to -\infty}f(x) = -\infty \qquad \lim_{x \to +\infty}f(x) = +\infty$$

54.Since $f$ is continuous, these limits and the intermediate value theorem show that its range is $\mathbb{R}.$ The function is strictly increasing, so it is a bijection from $\mathbb{R}$ onto $\mathbb{R}.$ Its inverse is the power function with reciprocal exponent:

$$f^{-1}(x) = x^{3/5}$$

55.Since $x^{5/3}/x = x^{2/3}$ tends to $+\infty$ as $x \to \pm\infty,$ the graph has no oblique asymptote. The endpoint limits above also rule out horizontal asymptotes.