EVALUATING FUNCTIONS FOR THE GIVEN VALUE OF X

Question 1 :

If f(x)  =  (x + 2)/(x² + 1) For every real number x. Evaluate the following expression.

(i)  f(2a)    (ii)  f(2a - 1)    (iii)  f(x² + 1)    (iv)  f(2x² + 3)

Solution :

(i)  f(2a)

f(x)  =  (x + 2)/(x² + 1)

x  =  2a

f(2a)  =  (2a + 2)/((2a)² + 1)

  =  (2a + 2)/(4a² + 1)

  =  2(a + 1)/(4a² + 1)

(ii)  f(2a - 1) 

f(x)  =  (x + 2)/(x² + 1)

x  =  2a - 1

f(2a - 1)  =  (2a - 1 + 2)/((2a - 1)² + 1)

  =  (2a + 1)/(4a² + 1 + 4a + 1)

  =  (2a + 1)/(4a² + 4a + 2)

(iii)  f(x² + 1)   

f(x)  =  (x + 2)/(x² + 1)

x  =  x² + 1

f(x² + 1)  =  (x² + 1 + 2)/((x² + 1)² + 1)

=  (x² + 3)/(x⁴ + 2x² + 1 + 1)

=  (x² + 3)/(x⁴ + 2x² + 2)

(iv)  f(2x² + 3)

f(x)  =  (x + 2)/(x² + 1)

x  =  2x² + 3

f(2x² + 3)  =  (2x² + 3 + 2)/((2x² + 3)² + 1)

=  (2x² + 5)/(4x⁴ + 12x² + 9 + 1)

=  (2x² + 5)/(4x⁴ + 12x² + 10)

Question 2 :

If g(x)  =  (x - 1)/(x + 2) For every real number x. Evaluate the following expression.

(i)  Find a number "b" such that g(b) = 4.

(ii)  Find a number b such that g(b) = 3

(iii) Evaluate and simplify the expression [g(x)−g(3)]/(x−3)

Solution :

(i)  Find a number "b" such that g(b) = 4.

 g(x)  =  (x - 1)/(x + 2)

 g(b)  =  (b - 1)/(b + 2)

 4  =  (b - 1)/(b + 2)

4(b + 2)  =  b - 1

4b + 8  =  b - 1

4b - b  =  -1 - 8

3b  =  -9

b  =  -3

(ii)  Find a number b such that g(b) = 3

 g(x)  =  (x - 1)/(x + 2)

 g(b)  =  (b - 1)/(b + 2)

3  =  (b - 1)/(b + 2)

3(b + 2)  =  b - 1

3b + 6  =  b - 1

3b - b  =  -1 - 6

2b  =  -7

b  =  -7/2

(iii) Evaluate and simplify the expression [g(x)−g(3)]/(x−3)

g(x)  =  (x - 1)/(x + 2)

 g(3)  =  (3 - 1)/(3 + 2)  =  2/5

[g(x)−g(3)]/(x−3)  =  [(x - 1)/(x + 2) - (2/5)]/(x−3) 

=  [5(x - 1) - 2(x + 2)/5(x + 2)]/(x−3) 

=  [(5x - 5 - 2x - 4)/5(x + 2)]/(x−3) 

=  [(3x - 9)/5(x + 2)]/(x−3) 

=  [3 (x - 3)/5(x + 2)]/(x−3) 

=  3/5(x + 2) 

Question 3 :

Assume that f is the function defined by

(i)  Evaluate f(2).

(ii)  Evaluate f(−3).

(iii)  Evaluate f(|x| + 1).

(iv)  Evaluate f(|x − 5| + 2).

Solution :

(i)  f(2)

Here the value of x is 2 which is greater than 0. So we have to choose the function f(x)  =  3x - 10

f(2)  =  3(2) - 10  =  6 - 10

  f(2)  =  -4

(ii)  Evaluate f(−3).

Here the value of x is 2 which is lesser than 0. So we have to choose the function f(x)  =  2x + 9

f(-3)  =  2(-3) - 9  =  - 6 - 9

  f(-3)  =  -15

(iii)  Evaluate f(|x| + 1)

f(|x| + 1)  =  3(|x|+ 1) - 10

=  3|x| + 3 - 10

f(|x| + 1)  =  3|x| - 10

(iv)  Evaluate f(|x − 5| + 2).

f(|x - 5| + 2)  =  3(|x - 5|+ 2) - 10

=  3|x - 5| + 6 - 10

f(|x - 5| + 2)  =  3|x - 5| - 4

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