Determine

Question

(a) f(g(x)) and the domain of the composite function, (b) g(f(x)) and the domain of the composite function?

67. f(x) = 3x +7 
g(x) = (x-7)/3 

68. f(x) = sqrt(x+1) 
g(x) = x^2-1 

72. The absolute value function is defined as follows: 

..........{-x....x<0.............. 
y= |x|={........................... 
..........{x.....x>and= to 0.. 

74. Sketch the graph of y = |2x-3|

Answer 1

(68). (a).

The functions are f(x) = √(x + 1) and g(x) = x² - 1.

f[g(x)] = f[x² - 1]                       [ Since g(x) = x² - 1]

           = √[(x² - 1) + 1]            [ Since f(x) = √(x + 1)]

           = x.

First find the domain of the inside function g(x) = x² - 1 :

The function g(x) = x² - 1 is a quadratic function or parabola function. There are no rational or radical expressions, so there is nothing that will restrict the domain. Any real number can be used for x to get a meaningful output.

The domain of g(x) = x² - 1 is (- ∞, ∞).

Next find the domain of the composite function f[g(x)] = x :

The function f[g(x)] = x is a linear function. There are no rational or radical expressions, so there is nothing that will restrict the domain. Any real number can be used for x to get a meaningful output.

The domain of f[g(x)] = x is (- ∞, ∞).

 

(68). (b).

The functions are f(x) = √(x + 1) and g(x) = x² - 1.

g[f(x)] = g[√(x + 1)]                       [ Since f(x) = √(x + 1)]

           = [√(x + 1)]² - 1                  [ Since g(x) = x² - 1]

           = x.

First find the domain of the inside function f(x) = √(x + 1) :

The domain of a function is all values of x, those makes the function mathematically correct.

Since there shouldn't be any negative numbers in the square root.

The expression (x + 1) should be zero or positive.

x + 1 ≥ 0

x ≥ - 1.

The domain of f(x) = √(x + 1) is [- 1, ∞).

Next find the domain of the composite function g[f(x)] = x :

The function g[f(x)] = x is a linear function. There are no rational or radical expressions, so there is nothing that will restrict the domain. Any real number can be used for x to get a meaningful output.

The domain of g[f(x)] = x is all real numbers, but you must keep the domain of the inside function. So the domain for the composite function is also [- 1, ∞).

Answer 2

(67). (a).

The functions are f(x) = 3x + 7 and g(x) = (x - 7)/3.

f[g(x)] = f[(x - 7)/3]                       [ Since g(x) = (x - 7)/3]

           = 3[(x - 7)/3] + 7               [ Since f(x) = 3x + 7]

           = x.

First find the domain of the inside function g(x) = (x - 7)/3 :

The function g(x) = (x - 7)/3 is a linear function . There are no rational or radical expressions, so there is nothing that will restrict the domain. Any real number can be used for x to get a meaningful output.

The domain of g(x) = (x - 7)/3 is (- ∞, ∞).

Next find the domain of the composite function f[g(x)] = x :

The function f[g(x)] = x is a linear function. There are no rational or radical expressions, so there is nothing that will restrict the domain. Any real number can be used for x to get a meaningful output.

The domain of f[g(x)] = x is (- ∞, ∞).

 

(67). (b).

The functions are f(x) = 3x + 7 and g(x) = (x - 7)/3.

g[f(x)] = g[3x + 7]                       [ Since f(x) = 3x + 7]

           = [(3x + 7) - 7]/3             [ Since g(x) = (x - 7)/3]

           = x.

First find the domain of the inside function f(x) = 3x + 7 :

The function f(x) = 3x + 7 is a linear function. There are no rational or radical expressions, so there is nothing that will restrict the domain. Any real number can be used for x to get a meaningful output.

The domain of 3x + 7 is (- ∞, ∞).

Next find the domain of the composite function g[f(x)] = x :

The function g[f(x)] = x is a linear function. There are no rational or radical expressions, so there is nothing that will restrict the domain. Any real number can be used for x to get a meaningful output.

The domain of g[f(x)] = x is (- ∞, ∞).

 

Answer 3

72.

The domain of the above function is (- ∞, ∞) and its graph is

Answer 4

74.

The absolute value function is y = | 2x - 3|.

If (2x - 3) ≥ 0 ⇒ x ≥ 3/2 then y = 2x - 3.

If (2x - 3) < 0 ⇒ x < 3/2 then y = - (2x - 3) = - 2x + 3.

Write the absolute value function in piecewise function.

The domain of the above function is (- ∞, ∞) and its graph is