applications and use of the inverse functions

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Applications and Use of the InverseFunctions

Examples on how to aplly and use inverse functions in real life situations andsolve problems in mathematics.

Example 1: Use inverse functions to solve equations.

Solve the following equation

Log ( x - 3) = 2

Solution to example 1:

• Since logarithmic and exponential functions areinverses of each other, we can write the following.

A = Log (B) if and only B = 10 A

• Use the above property of logarithmic andexponential functions to rewite the given equation asfollows.

x - 3 = 10 2

• Solve for x to obtain.

x = 103

Example 2: Use inverse functions to find range of functions.

Find the range of function f give by

f(x) = 2 x / (x - 3)


• We know that the range of a one to one function isthe domain of its inverse. Let us first show thatfunction f given above is a one to one function. Start

with .

f(a) = f(b)

2 a / (a - 3) = 2 b / (b - 3)

• Multiply all terms of the above equation by (a - 3)(b -3) and simplify to obtain.



2a (b - 3) = 2 b(a - 3)

• Expand.

2a b - 6 a = 2 ba - 6 b

• Add - 2 a b to both sides and simplify to obtain.

a = b

• Hence the given function is a one to one. let us findits inverse.

y = 2 x / (x - 3)

• Interchange x and y and solve for y.

x = 2 y / (y - 3)

y = 3x / (2 - x)

• The inverse of function f is given by.

f -1 (x) = 3x / (2 - x)

• The domain of f -1 is the set of all real values except x= 2. Hence the range of f is the set of all real valuesexcept 2.

Example 3: Use inverse functions find the angle of elevationof a camera.

A camera is to take a series of photographs of a hot air balloon rising vertically. The distance between the cameraat (B) and the launching point of the balloon (A) is 300meters. The camera must keep the balloon on sight andtherefore its angle of elevation t must change with the heightx of the balloon.

a) Find angle t as a function of the height x.

b) Find angle t in degrees when x is equal to 150, 300 and600 meters. (approximate your answer to 1 decimal place).

c) Graph t as a function of x.




• a) The opposite and the adjacent sides to angle t arex and 300 meters respectively, hence.

tan(t) = x / 300

• We now use the property of the tangent function andits inverse.

tan -1(tan(x)) = x

• To rewrite the equation tan(t) = x / 300 as follows .

tan -1(tan(t)) = tan -1( x / 300 )

• Simplify the left side of the above equation to obtain .

t = tan -1( x / 300 )

• b) The values of t at 150, 300 and 600 are foundusing a calculator..

t(150) = 25.6 degrees (approximated to 1 decimalplace)

t(300) = 45.0 degrees


• c) We use the values of t in part b) and extra pointsand graph t as a function of x..

x t



0 0

150 25.6

300 45.0

600 63.4

1200 76.0

3000 84.3

•

•

Example 4: Use inverse functions to find radius of rightcircular cone.

Five right circular cones, with the same height h = 50 cm,are to be constructed. The volumes of these cones are to be200, 400, 800, 1600 and 3200 cm3. Find the radius of thebase of each cone.


•

The formula of the volume V of a right circular conewith height h and radius r is given by.

V = (1/3) pi r 2 h

• Since we need to find the radius, we need to solvethe above eqaution for r to obtain.



r = SQRT(3 V / pi h)

• What we have done above is to find the inversefunction of V. We do not need to interchange thevariables V and r because they have different

meaning in this problem..

• We now calculate the radius for each cone using theformula for the radius above..

a) V = 200 , r = SQRT(3*200 / 50*pi) = 1.95 cm

b) V = 400 , r = SQRT(3*400 / 50*pi) = 2.76 cm

c) V = 800 , r = SQRT(3*800 / 50*pi) = 3.91 cm

d) V = 1600 , r = SQRT(3*1600 / 50*pi) = 5.53 cm

e) V = 3200 , r = SQRT(3*3200 / 50*pi) = 7.82 cm

Example 5: Use inverse functions to solve populationproblems.

The population of a certain city increase according to thefollowing formula

P = 200,000 e 0.01 t

where P is the population and t the number of years, with t =0 corresponding to the year 2000.

When will the population be 300,000, 400,000 and 500,000?


• We need to find t first by solving the given formula for t. Divide both sides of the given formula by 200,000..

P / 200,000 = e 0.01 t

• We now use the fact that the exponential andlogarithmic functions are inverses of each other andrewrite the above exponential expression as follows.

0.01 t = ln ( P / 200,000 )

• Solve for t..



t = ln ( P / 200,000 ) / 0.01

• We now find t for the different values of P givenabove..

a) P = 300,000 , t = ln ( 300,000 / 200,000 ) / 0.01 =40.55 years , year 2041

a) P = 400,000 , t = ln ( 400,000 / 200,000 ) / 0.01 =69.31 years , year 2070

a) P = 500,000 , t = ln ( 500,000 / 200,000 ) / 0.01 =91.63 years , year 2092

More links and references related to the inverse functions.

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Updated: 27 November 2007 (A Dendane)

Applications and Use of the InverseFunctions

Examples on how to aplly and use inverse functions in real life situations andsolve problems in mathematics.

Example 1: Use inverse functions to solve equations.

Solve the following equation

Log ( x - 3) = 2


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• Since logarithmic and exponential functions areinverses of each other, we can write the following.

A = Log (B) if and only B = 10 A

• Use the above property of logarithmic andexponential functions to rewite the given equation asfollows.

x - 3 = 10 2

• Solve for x to obtain.

x = 103

Example 2: Use inverse functions to find range of functions.

Find the range of function f give by

f(x) = 2 x / (x - 3)


• We know that the range of a one to one function isthe domain of its inverse. Let us first show thatfunction f given above is a one to one function. Startwith .

f(a) = f(b)

2 a / (a - 3) = 2 b / (b - 3)

• Multiply all terms of the above equation by (a - 3)(b -3) and simplify to obtain.

2a (b - 3) = 2 b(a - 3)

• Expand.

2a b - 6 a = 2 ba - 6 b

• Add - 2 a b to both sides and simplify to obtain.

a = b

• Hence the given function is a one to one. let us findits inverse.

y = 2 x / (x - 3)



• Interchange x and y and solve for y.

x = 2 y / (y - 3)

y = 3x / (2 - x)

• The inverse of function f is given by.

f -1 (x) = 3x / (2 - x)

• The domain of f -1 is the set of all real values except x= 2. Hence the range of f is the set of all real valuesexcept 2.

Example 3: Use inverse functions find the angle of elevationof a camera.

A camera is to take a series of photographs of a hot air balloon rising vertically. The distance between the cameraat (B) and the launching point of the balloon (A) is 300meters. The camera must keep the balloon on sight andtherefore its angle of elevation t must change with the heightx of the balloon.

a) Find angle t as a function of the height x.

b) Find angle t in degrees when x is equal to 150, 300 and

600 meters. (approximate your answer to 1 decimal place).

c) Graph t as a function of x.


• a) The opposite and the adjacent sides to angle t are



x and 300 meters respectively, hence.

tan(t) = x / 300

• We now use the property of the tangent function and

its inverse.

tan -1(tan(x)) = x

• To rewrite the equation tan(t) = x / 300 as follows .

tan -1(tan(t)) = tan -1( x / 300 )

• Simplify the left side of the above equation to obtain .

t = tan -1( x / 300 )

• b) The values of t at 150, 300 and 600 are foundusing a calculator..


t(300) = 45.0 degrees


•

c) We use the values of t in part b) and extra pointsand graph t as a function of x..

x t

0 0

150 25.6

300 45.0

600 63.4

1200 76.0

3000 84.3

•



•

Example 4: Use inverse functions to find radius of rightcircular cone.

Five right circular cones, with the same height h = 50 cm,are to be constructed. The volumes of these cones are to be200, 400, 800, 1600 and 3200 cm3. Find the radius of the

base of each cone.Solution to example 4:

• The formula of the volume V of a right circular conewith height h and radius r is given by.

V = (1/3) pi r 2 h

• Since we need to find the radius, we need to solvethe above eqaution for r to obtain.

r = SQRT(3 V / pi h)

• What we have done above is to find the inversefunction of V. We do not need to interchange thevariables V and r because they have differentmeaning in this problem..

• We now calculate the radius for each cone using theformula for the radius above..



a) V = 200 , r = SQRT(3*200 / 50*pi) = 1.95 cm

b) V = 400 , r = SQRT(3*400 / 50*pi) = 2.76 cm

c) V = 800 , r = SQRT(3*800 / 50*pi) = 3.91 cm

d) V = 1600 , r = SQRT(3*1600 / 50*pi) = 5.53 cm

e) V = 3200 , r = SQRT(3*3200 / 50*pi) = 7.82 cm

Example 5: Use inverse functions to solve populationproblems.

The population of a certain city increase according to thefollowing formula

P = 200,000 e 0.01 t

where P is the population and t the number of years, with t =0 corresponding to the year 2000.

When will the population be 300,000, 400,000 and 500,000?


• We need to find t first by solving the given formula for t. Divide both sides of the given formula by 200,000..

P / 200,000 = e 0.01 t

• We now use the fact that the exponential andlogarithmic functions are inverses of each other andrewrite the above exponential expression as follows.

0.01 t = ln ( P / 200,000 )

• Solve for t..

t = ln ( P / 200,000 ) / 0.01• We now find t for the different values of P given

above..

a) P = 300,000 , t = ln ( 300,000 / 200,000 ) / 0.01 =40.55 years , year 2041

a) P = 400,000 , t = ln ( 400,000 / 200,000 ) / 0.01 =



69.31 years , year 2070

a) P = 500,000 , t = ln ( 500,000 / 200,000 ) / 0.01 =91.63 years , year 2092

More links and references related to the inverse functions.





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e2 = - i1 R3 + i2(R2 + R3)

and then write it in matrixform as follows

The above is a matrixequation that may besolved using any knownmethod to solve systems of

equations. Let e, R and i bematrices given by

The solution to the above

matrix equation is given by

where R -1 is the inversematrix of R and is given by.

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