Algebraic Functions; Variation

Definition of Algebraic Function

An algebraic function is any function for which the rule is a polynomial or can be derived from polynomials by addition, subtraction, multiplication, division, or raising to an integer or rational exponent.

EXAMPLE 16.1 Examples of algebraic functions include:

  1. Polynomial functions, such as $ f(x) = 5x^{2} - 3x $

  2. Rational functions, such as $ f(x) = 12/x^{2} $

  3. Absolute value functions, such as $ f(x) = |x - 3| $ , since $ |x - 3| = $

  4. Other functions involving rational powers, such as $ f(x) = $ , $ f(x) = $ , $ f(x) = 1/ $ , $ f(x) = $ , and so on

V ariation

The term variation is used to describe many forms of simple functional dependence. The general pattern is that one variable, called the dependent variable, is said to vary as a result of changes in one or more other variables, called the independent variables. Variation statements always include a nonzero constant multiple, referred to as the constant of variation, or constant of proportionality, and often denoted k.

Direct Variation

To describe a relation of the form \(y = kx\), the following language is used:

  1. y varies directly as x (occasionally, y varies as x).

  2. y is directly proportional to x.

EXAMPLE 16.2 Given that p varies directly as q, find an expression for p in terms of q if p = 300 when q = 12.

  1. Since p varies directly as q, write p = kq.

  2. Since p = 300 when q = 12, substitute these values to obtain $ 300 = k(12) $ , or k = 25.

  3. Hence p = 25q is the required expression.

I nverse Variation

To describe a relation of the form xy = k, or y = k/x, the following language is used:

  1. y varies inversely as x.

  2. y is inversely proportional to x.

EXAMPLE 16.3 Given that s varies inversely as t, find an expression for s in terms of t if s = 5 when t = 8.

  1. Since s varies inversely as t, write s = k/t.

  2. Since s = 5 when t = 8, substitute these values to obtain 5 = k/8, or k = 40.

  3. Hence s = 40/t is the required expression.

Joint Variation

To describe a relation of the form z = kxy, the following language is used:

  1. z varies jointly as x and y.

  2. z varies directly as the product of x and y.

EXAMPLE 16.4 Given that z varies jointly as x and y and z = 3 when x = 4 and y = 5, find an expression for z in terms of x and y.

  1. Since z varies jointly as x and y, write z = kxy.

  2. Since z = 3 when x = 4 and y = 5, substitute these values to obtain $ 3 = k $ , or $ k = $ .

  3. Hence $ z = xy $ .

Combined Variation

These types of variation can also be combined.

EXAMPLE 16.5 Given that z varies directly as the square of x and inversely as y and z = 5 when x = 3 and y = 12, find an expression for z in terms of x and y.

  1. Write $ z = $ .

  2. Since z = 5 when x = 3 and y = 12, substitute these values to obtain $ 5 = k $ or k = 20/3.

  3. Hence $ z = . $

SOLVED PROBLEMS

16.1. State the domain and range, and sketch a graph for:

  1. Domain $ [0, ) $

Range: $ [0, ) $

The graph is shown in Fig. 16-1.

  1. Domain: R

Range: R

The graph is shown in Fig. 16-2.

Figure 16-2
  1. Domain: $ [0, ) $

Range: $ [0, ) $

The graph is shown in Fig. 16-3.

  1. Domain: R

Range: R

The graph is shown in Fig. 16-4.

16.2. State the domain and range, and sketch a graph for:

  1. $ f(x) = 1/ $ (b) $ f(x) = 1/ $

  2. Domain: $ (0, ) $

Range: $ (0, ) $

The graph is shown in Fig. 16-5.

  1. Domain: $ {x R | x } $

Range: $ {y R | y } $

The graph is shown in Fig. 16-6.

Figure 16-4

Figure 16-6

16.3. Analyze and sketch a graph for (a) $ f(x)= $ ; (b) $ f(x)=- $ .

  1. If $ y = $ , then $ x^{2} + y^{2} = 9 $ , $ y $ . Thus, the graph of the function is the upper half (semicircle) of the graph of $ x^{2} + y^{2} = 9 $ .

The domain is $ {x R x } $ and the range is $ {y R y } $ .

The graph is shown in Fig. 16-7.

  1. If $ y = - $ , then $ x^{2} + y^{2} = 9 $ , $ y $ . Thus, the graph of the function is the lower half (semicircle) of the graph of $ x^{2} + y^{2} = 9 $ .

The domain is $ {x R x } $ and the range is $ {y R y } $ . The graph is shown in Fig. 16-8.

Figure 16-8

16.4. If s varies directly as the square of x and s = 5 when x = 4, find s when x = 20.

  1. Since s varies directly as the square of x, write $ s = kx^{2} $ .

  2. Since s = 5 when x = 4, substitute these values to obtain $ 5 = k ^{2} $ , or $ k = $ .

  3. Hence $ s = 5x^{2}/16 $ . Thus, when x = 20, $ s = 5(20)^{2}/16 = 125 $ .

16.5. If y is directly proportional to the cube root of x and y = 12 when x = 64, find y when $ x = $ .

  1. Since y is directly proportional to the cube root of x, write $ y = k $ .

  2. Since y = 12 when x = 64, substitute these values to obtain $ 12 = k = 4k $ , or k = 3.

  3. Hence $ y = 3 $ . Thus, when $ x = $ , $ y = 3 = $ .

16.6. If I is inversely proportional to the square of t, and I = 100 when t = 15, find I when t = 12.

  1. Since I is inversely proportional to the square of t, write $ I = $ .

  2. Since I = 100 when t = 15, substitute these values to obtain $ 100 = $ , or k = 22,500.

  3. Hence $ I = $ . Thus, when t = 12, $ I = = 156.25 $ .

16.7. If u varies inversely as the cube root of x, and u = 56 when x = -8, find u when x = 1000.

  1. Since u varies inversely as the cube root of x, write $ u = $ .

  2. Since u = 56 when x = -8, substitute these values to obtain $ 56 = $ , or k = -112.

  3. Hence $ u = $ . Thus, when x = 1000, $ u = = -11.2 $ .

16.8. If z varies jointly as x and y, and z = 3 when x = 4 and y = 6, find z when x = 20 and y = 9.

  1. Since z varies jointly as x and y, write z = kxy.

  2. Since z = 3 when x = 4 and y = 6, substitute these values to obtain $ 3 = k $ or $ k = $ .

  3. Hence \(z = xy/8\). Thus, when \(x = 20\) and \(y = 9\), \(z = (20 \cdot 9)/8 = 22.5\).

16.9. If P varies jointly as the square of x and the fourth root of y, and P = 24 when x = 12 and y = 81, find P when x = 1200 and $ y = $ .

  1. Since P varies jointly as the square of x and the fourth root of y, write $ P = kx^{2} $ .

  2. Since P = 24 when x = 12 and y = 81, substitute these values to obtain $ 24 = k ^{2} $ or $ k = $ .

  3. Hence $ P = $ . Thus, when x = 1200 and $ y = $ , $ P = = 40,000 $ .

16.10. Hooke’s law states that the force F needed to stretch a spring x units beyond its natural length is directly proportional to x. If a certain spring is stretched 0.5 inches from its natural length by a force of 6 pounds, find the force necessary to stretch the spring 2.25 inches.

  1. Since F is directly proportional to x, write F = kx.

  2. Since \(F = 6\) when \(x = 0.5\), substitute these values to obtain \(6 = k(0.5)\) or \(k = 12\).

  3. Hence \(F = 12x\). Thus, when \(x = 2.25\), \(F = 12(2.25) = 27\) pounds.

16.11. Ohm’s law states that the current I in a direct-current circuit varies inversely as the resistance R. If a resistance of 12 ohms produces a current of 3.5 amperes, find the current when the resistance is 2.4 ohms.

  1. Since I varies inversely as R, write I = k/R.

  2. Since \(I = 3.5\) when \(R = 12\), substitute these values to obtain \(3.5 = k/12\) or \(k = 42\).

  3. Hence I = 42/R. Thus, when R = 2.4, I = 42/2.4 = 17.5 amperes.

16.12. The pressure P of wind on a wall varies jointly as the area A of the wall and the square of the velocity v of the wind. If P = 100 pounds when A = 80 square feet and v = 40 miles per hour, find P if A = 120 square feet and v = 50 miles per hour.

  1. Since P varies jointly as A and v, write $ P = kAv^{2} $ .

  2. Since P = 100 when A = 80 and v = 40, substitute these values to obtain $ 100 = k ^{2} $ or k = 1/1280.

  3. Hence $ P = A v^{2}/1280 $ . Thus, when A = 120 and v = 50, $ P = 120 ^{2}/1280 = 234.375 $ pounds.

16.13. The weight w of an object on or above the surface of the earth varies inversely as the square of the distance d of the object from the center of the earth. If an astronaut weighs 120 pounds at the surface of the earth, how much (to the nearest pound) would she weigh in a satellite 400 miles above the surface? (Use 4000 miles as the radius of the earth.)

  1. Since w varies inversely as the square of d, write $ w = k/d^{2} $ .

  2. Since w = 120 at the surface of the earth, when d = 4000, substitute these values to obtain $ 120 = k/4000^{2} $ , or $ k = 1.92 ^{9} $ .

  3. Hence $ w = 1.92 {9}/d{2} $ . Thus, when $ d = 4000 + 400 = 4400 $ , $ w = 1.92 {9}/4400{2} $ , or approximately 99 pounds.

16.14. The volume V of a given mass of gas varies directly as the temperature T and inversely as the pressure P. If a gas has volume 16 cubic inches when the temperature is $ 320^{} $ K and the pressure is 300 pounds per square inch, find the volume when the temperature is $ 350^{} $ K and the pressure is 280 pounds per square inch.

  1. Since V varies directly as T and inversely as P, write $ V = kT/P $ .

  2. Since V = 16 when T = 320 and P = 300, substitute these values to obtain $ 16 = k /300 $ or k = 15.

  3. Hence V = 15T/P. Thus, when T = 350 and P = 280, $ V = 15 /280 = 18.75 $ cubic inches.

16.15. If y varies directly as the square of x, what is the effect on y of doubling x?

  1. Since $ y = kx^{2} $ , write $ k = y/x^{2} $ .

  2. While x and y vary, k remains constant; hence for different x and y values, $ y_{1}/x_{1}^{2} = y_{2}/x_{2}^{2} $ , or $ y_{2} = y_{1}x_{2}{2}/x_{1}{2} $ .

  3. Hence, if $ x_{2}=2x_{1} $ , $ y_{2}=y_{1}(2x_{1}){2}/x_{1}{2}=4y_{1} $ . Thus, if x is doubled, y is multiplied by 4.

16.16. If y varies inversely as the cube of x, what is the effect on y of doubling x?

  1. Since $ y = k/x^{3} $ , write $ k = x^{3}y $ .

  2. While x and y vary, k remains constant; hence for different x and y values, $ x_{1}^{3}y_{1} = x_{2}^{3}y_{2} $ , or $ y_{2} = y_{1}x_{1}{3}/x_{2}{3} $ .

  3. Hence, if $ x_{2}=2x_{1} $ , $ y_{2}=y_{1}x_{1}{3}/(2x_{1}){3}=y_{1}/8 $ . Thus, if x is doubled, y is divided by 8.

16.17. The strength W of a rectangular beam of wood varies jointly as the width w and the square of the depth d, and inversely as the length L of the beam. What would be the effect on W of doubling w and d while decreasing L by a factor of 20%?

  1. Since $ W = kwd^{2}/L $ , write $ k = WL/(wd^{2}) $ .

  2. For different values of the variables, k remains constant, hence W1L1/(w1d12) = W2L2/(w2d2).

  3. Hence, if $ w_{2}=2w_{1}, d_{2}=2d_{1} $ , and $ L_{2}=L_{1}-0.2L_{1}=0.8L_{1} $ , write:

$ = $ and solve for $ W_{2} $ to obtain $ W_{2}==10W_{1} $ . Thus, W would be multiplied by 10.

SUPPLEMENTARY PROBLEMS

16.18. State the domain and range and sketch a graph of the following functions:

\[ (a)f(x)=\sqrt[3]{x-2} \]

\[ (b)f(x)=-1/\sqrt{x+3} \]

\[ (c)f(x)=\sqrt{4-(x+2)^{2}} \]

Ans. (a) Domain: R, Range R

The graph is shown in Fig. 16-9.

Figure 16-9
  1. Domain: $ {x R x > -3} $

Range: $ {y R y < 0} $

The graph is shown in Fig. 16-10.

Figure 16-10
  1. Domain: \(\{x \in \mathbb{R} \mid -4 \leq x \leq 0\}\)

Range: \(\{y \in \mathbb{R} \mid 0 \leq y \leq 2\}\)

The graph is shown in Fig. 16-11.

Figure 16-11

16.19. If y varies directly as the fourth power of x, and y = 2 when $ x = $ , find y when x = 2.

Ans. 512

16.20. If y varies inversely as the square root of x, and y = 2 when $ x = $ , find y when x = 2.

16.21. If a spring of natural length 5 centimeters is displaced 0.3 centimeter from its natural length by a weight of 6 pounds, use Hooke’s law (Problem 16-10) to determine the weight necessary to displace the spring 1 centimeter.

Ans. 20 pounds

16.22. Newton’s law of cooling states that the rate r at which a body cools is directly proportional to the difference between the temperature T of the body and the temperature $ T_{0} $ of its surroundings. If a cup of hot coffee at temperature $ 140^{} $ is in a room at temperature $ 68^{} $ and is cooling at the rate of $ 9^{} $ per minute, find the rate at which it will be cooling when its temperature has dropped to $ 116^{} $ .

Ans. $ 6^{} $ per minute.

16.23. Kepler’s third law states that the square of the time T required for a planet to complete one orbit around the sun (the period, that is, the length of one planetary year) is directly proportional to the cube of the average distance d of the planet from the sun. For the planet Earth, assume $ d = 93 ^{6} $ miles and T = 365 days. Find (a) the period of Mars, given that Mars is approximately 1.5 times as distant from the sun as Earth; (b) the average distance of Venus from the sun, given that the period of Venus is approximately 223 Earth days.

Ans. (a) 671 Earth days; (b) $ 67 ^{6} $ miles

16.24. The resistance R of a wire varies directly as the length L and inversely as the square of the diameter d. A 4-meter-long piece of wire with a diameter of 6 millimeters has a resistance of 600 ohms. What diameter should be used if a 5-meter piece of this wire is to have a resistance of 1000 ohms?

Ans. $ $ millimeters

16.25. Coulomb’s law states that the force F of attraction between two oppositely charged particles varies jointly as the magnitudes $ q_{1} $ and $ q_{2} $ of their electrical charges and inversely as the square of the distance d between the particles. What is the effect on F of doubling the magnitude of the charges and halving the distance between them?

Ans. The force is multiplied by a factor of 16.