Several questions on functions are presented and their detailed solutions discussed. The questions cover a wide range of concepts related to functions such as definition, domain, range, evaluation, composition and transformations of the graphs of functions.
Question 1: Is the graph shown below that of a function?
Solution to Question 1:

Vertical line test: A vertcal line at x = 0 for example cuts the graph at two points. The graph is not that of a function.
Question 2: Does the equation
y^{ 2} + x = 1
represents a function y in terms of x?
Solution to Question 2:

Solve the above equation for y
y^{ 2}= 1  x
y = + SQRT(1  x) or y =  SQRT(1  x)

For one value of x we have two values of y and this is not a function.
Question 3: Function f is defined by
f(x) =  2 x^{ 2} + 6 x  3
find f( 2).
Solution to Question 3:

Substitute x by 2 in the formula of the function and calculate f(2) as follows
f(2) =  2 (2)^{ 2} + 6 (2)  3
f(2) = 23
Question 4: Function h is defined by
h(x) = 3 x^{ 2}  7 x  5
find h(x  2).
Solution to Question 4:

Substitute x by x  2 in the formula of function h
h(x  2) = 3 (x  2)^{ 2}  7 (x  2)  5

Expand and group like terms
h(x  2) = 3 ( x ^{ 2}  4 x + 4 )  7 x + 14  5
= 3 x ^{ 2}  19 x + 7
Question 5: Functions f and g are defined by
f(x) =  7 x  5 and g(x) = 10 x  12
find (f + g)(x)
Solution to Question 5:

(f + g)(x) is defined as follows
(f + g)(x) = f(x) + g(x) = ( 7 x  5) + (10 x  12)

Group like terms to obtain
(f + g)(x) = 3 x  17
Question 6: Functions f and g are defined by
f(x) = 1/x + 3x and g(x) = 1/x + 6x  4
find (f + g)(x) and its domain.
Solution to Question 6:

(f + g)(x) is defined as follows
(f + g)(x) = f(x) + g(x)
= (1/x + 3x) + (1/x + 6x  4)

Group like terms to obtain
(f + g)(x) = 9 x  4

The domain of function f + g is given by the intersection of the domains of f and g
Domain of f + g is given by the interval (infinity , 0) U (0 , + infinity)
Question 7: Functions f and g are defined by
f(x) = x^{ 2} 2 x + 1 and g(x) = (x  1)(x + 3)
find (f / g)(x) and its domain.
Solution to Question 7:

(f / g)(x) is defined as follows
(f / g)(x) = f(x) / g(x) = (x^{ 2} 2 x + 1) / [ (x  1)(x + 3) ]

Factor the numerator of f / g and simplify
(f / g)(x) = f(x) / g(x) = (x  1)^{ 2} / [ (x  1)(x + 3) ]
= (x  1) / (x + 3) , x not equal to 1

The domain of f / g is the intersections of the domain of f and g excluding all values of x that make the numerator equal to zero. The domain of f / g is given by
(infinity , 3) U (3 , 1) U (1 , + infinity)
Question 8: Find the domain of the real valued function h defined by
h(x) = SQRT ( x  2)
Solution to Question 8:

For function h to be real valued, the expression under the square root must be positive or equal to 0. Hence the condition
x  2 >= 0

Solve the above inequality to obtain the domain in inequality form
x >= 2

and interval form
[2 , + infinity)
Question 9: Find the domain of
g(x) = SQRT (  x^{ 2} + 9) + 1 / (x  1)
Solution to Question 9:

For a value of the variable x to be in the domain of function g given above, two conditions must be satisfied: The expression under the square root must not be negative
 x^{ 2} + 9 >= 0

and the denomirator of 1 / (x  1) must not be zero
x not equal to 1
or in interval form
(infinity , 1) U (1 , + infinity)

The solution to the inequality  x^{ 2} + 9 >= 0 is given by the interval
[3 , 3]

Since x must satisfy both conditions, the domain of g is the intersection of the sets
(infinity , 1) U (1 , + infinity) and [3 , 3]
[3 , 1) U (1 , +3]
Question 10: Find the range of
f(x) =  x  2  + 3
Solution to Question 10:

 x  2  is an absolute value and is either positive or equal to zero as x takes real values, hence
 x  2  >= 0

Add 3 to both sides of the above inequality to obtain
 x  2  + 3 >= 3

The expression on the left side of the above inequality is equal to f(x), hence
f(x) >= 3

The above inequality gives the range as the interval
[3 , + infinity)
Question 11: Find the range of
f(x) = x^{ 2}  10
Solution to Question 11:

x^{ 2} is either negative or equal to zero as x takes real values, hence
x^{ 2} <= 0

Add 10 to both sides of the above inequality to obtain
x^{ 2}  10 <= 10

The expression on the left side is equal to f(x), hence
f(x) <= 10

The above inequality gives the range of f as the interval
(infinity , 10]
Question 12: Find the range of
h(x) = x^{ 2}  4 x + 9
Solution to Question 12:

h(x) is a quadratic function, so let us first write it in vertex form using completing the square
h(x) = x^{ 2}  4 x + 9
= x^{ 2}  4 x + 4  4 + 9
= (x  2)^{ 2} + 5

(x  2)^{ 2} is either positive or equal to zero as x takes real values, hence
(x  2)^{ 2} >= 0

Add 5 to both sides of the above inequality to obtain
(x  2)^{ 2} + 5 >= 5

The above inequality gives the range of h as the interval
[5 , + infinity)
Question 13: Functions g and h are given by
g(x) = SQRT(x  1) and h(x) = x^{ 2} + 1
Find the composite function (g _{o} h)(x).
Solution to Question 13:

The definition of the absolute value gives
(g _{o} h)(x) = g(h(x))
= g(x^{ 2} + 1)
= g(x^{ 2} + 1)
= SQRT(x^{ 2} + 1  1)
=  x 

So
(g _{o} h)(x) =  x 
Question 14: How is the graph of f(x  2) compared to the graph of f(x)?
Solution to Question 14:

The graph of f(x  2) is that of f(x) shifted 2 units to the right.
Question 15: How is the graph of h(x + 2)  2 compared to the graph of h(x)?
Solution to Question 15:

The graph of h(x + 2)  2 is that of h(x) shifted 2 units to the left and 2 unit downward.
Question 16: Express the perimeter P of a square as a function of its area.
Solution to Question 16:

A square shape with side x has perimeter P given by
P = 4 x

and an area A given by
A = x^{ 2}

Solve the equation P = 4 x for x
x = P / 4

and substitute into the formula for A to obtain
A = (P / 4)^{ 2}
= P ^{ 2} / 16

For any square shape the area A may be expressed as a function the perimeter P as follows
A = P ^{ 2} / 16
Exercises:

Evaluate f(3) given that f(x) =  x  6  + x^{ 2}  1
 Find f(x + h)  f(x) given that f(x) = a x + b

Find the domain of f(x) = SQRT(x^{ 2}  x + 2)
 Find the range of g(x) =  SQRT( x + 2)  6

Find (f o g)(x) given that f(x) = SQRT(x) and g(x) = x^{ 2}  2x + 1
 How do you obtain the graph of  f(x  2) + 5 from the graph of f(x)?
Answers to Above Exercises:
 f(3) = 11
 f(x + h)  f(x) = a h
 [2 , 1]
 ( infinity ,  6]

(f _{o} g)(x) =  x  1 
 Shift the graph of f 2 units to the right then reflect it on the x axis, then shift it upward 5 units.
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