FINDING RANGE OF RATIONAL FUNCTIONS

The range of a real function of a real variable is the set of all real values taken by f(x) at points in its domain. In order to find the range of real function f(x), we may use the following steps.

Steps Involved in Finding Range of Rational Function :

By finding inverse function of the given function, we may easily find the range. In order to find the inverse function, we have to follow the steps given below.

(i)  Put y = f(x)

(ii)  Solve the equation y = f(x) for x in terms of y.

(iii)  By replacing x by y and y by x, we get inverse function.

(iv)  The  values that we get for the inverse function by applying the the domain is known as range.

Domain of f(x)  =  Range of f^-1(x)

Range of f(x)  =  Domain of f^-1(x)

Solved Questions

Question 1 :

Find the range of the function f(x) is given by

f(x)  =  (x - 2)/(3 - x)

Solution :

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

Let us solve for x, in terms of y.

(3 - x)y  = (x - 2)

3y - xy  =  x - 2

3y + 2  =  x + xy

x(1 + y)  =  (3y + 2)

x  =  (3y + 2)/(1 + y)

Inverse function :

f^-1(x)   =  (3x + 2)/(1 + x)

To find the possible values inverse function, we have to equate the denominator to zero.

1 + x  =  0

x  =  -1

The possible values of x of inverse function is all real values except -1.

Hence the range of f(x) is R - {-1}.

(ii)  Find the range of f(x)  =  √(16 - x²)

Solution :

y  =  √(16 - x²)

y²  =  16 - x²

Solving for x,

x²  =  16 - y²

x²  =  4² - y²

x²  =  4² - y²

x  =  √(4² - y²)

Inverse function :

f^-1(x)  =  √(4² - x²)

If the values of x is more than 5, then we will get negative values inside the radical sign.

The domain for the given function f(x) is [-4, 4]. By applying those values of x, we get the values between 0 to 4.

Hence the range is [0, 4].

(iii)  Find the domain and range of real valued function f(x) is given by f(x)  =  √[(x - 2)/(3 - x)]

Solution :

Let y  = √[(x - 2)/(3 - x)]

Domain :

3 - x  =  0

x  =  3

Domain is set of possible real values except 3

Hence the required domain is R - {3}

y²  =  (x - 2) / (3 - x)

y² (3 - x)  =  (x - 2)

3y² - xy²  =  x - 2

3y² + 2  =  x + xy²

x(1 + y²)  =  3y² + 2

x  =  (3y² + 2)/(1 + y²)

Inverse of the given function is :

y  =  (3y² + 2)/(1 + y²)

f^-1(x)  =  (3x² + 2)/(1 + x²)

By applying the domain values in this function, we get positive values for inverse function.

So, the range is [0, ∞)

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