College Algebra Exercises Section 2.2
55) What is the domain of the function f whose rule is given by the equation f(x) = 5x2 + 2x -1?
According to the accepted convention the domain of a function, unless otherwise specified, will be the largest set of real numbers for which the defining equation (the rule) is defined (makes sense). When the defining equation is a quadratic equation the domain is all real numbers
58) What is the
domain
of the
function
s whose
rule
is given by the equation
According to the accepted convention the domain of a function, unless
otherwise specified, will be the largest set of real numbers for which the
defining equation (the rule) is defined (makes sense). When the defining equation is a
rational equation the domain is all real numbers that do not cause the denominator
to be zero.(Sometimes other numbers
are excluded from the domain because of conditions in the numerator.)The only number that will cause the
denominator to be zero in the function s, is –5. Therefore, the domain of this function s is all real numbers
except –5. The easiest way to represent
this set is to use set builder notation and write 
60) What is the
domain
of the
function
f whose
rule
is given by 
According to the accepted convention the domain of a function, unless otherwise specified, will be the largest set of real numbers for which the defining equation (the rule) is defined (makes sense). When the defining equation is an odd root of an expression, the equation makes sense for all real numbers. Therefore the domain of the function f is the set of all real numbers.
62) What is the
domain
of the
function
f whose
rule
is given by the equation
According to the accepted convention the domain of a function, unless
otherwise specified, will be the largest set of real numbers for which the
defining equation is defined (makes sense).
When the defining equation is a rational equation the domain
is all real numbers that do not cause the denominator to be zero.
To determine the numbers which cause a zero
in the denominator we must solve the equation x(x – 2) = x2 – 2x =
0.
The solutions are clearly x = 0 and
x = 2.
Therefore the domain of the
original function h is all real numbers R except 0 and 2.
We can write this with set builder notation as 
We can write this with interval notation as 
68) What is the
domain
of the
function
f whose
rule
is given by the equation
According to the accepted convention the domain of a function, unless
otherwise specified, will be the largest set of real numbers for which the
defining equation is defined (makes sense).
When the defining equation involves even roots, then all
numbers which cause a negative number under the radical sign must be
excluded.
When the defining equation is
a rational equation the domain is all real numbers that do not cause the
denominator to be zero.
Note that this
function involves both even roots and rational expressions.
The domain is therefore all real numbers R
for which (x – 3)(x + 3) = x2
– 9 > 0.
This inequality is true
when the graph of y = x2 – 9 is above the x-axis.
The graph of y = x2 – 9 is a parabola which
opens up and crosses the x-axis at –3 and 3.
It is therefore above the x-axis when x < -3 or x > 3.
The domain of the original function f is therefore {x | x
< -3 or x > 3}.
Using interval notation we would write
69) Consider the function f whose domain is the set {-2, -1, 0, 1, 2}and whose rule is f(x) = x2
Determine the range of f and determine the set of ordered pairs which constitute the graph of f.
Solution: The range will be the set {f(-2), f(-1), f(0), f(1), f(2)}= {4, 1, 0}
The set of ordered pairs is {(-2, 4), (-1, 1), (0, 0), (1, 1), (2, 4)}
Here is a picture of the graph of this function.
70) Consider the function f whose domain is the set {-2, -1, 0, 1, 2}and whose rule is f(x) = x2- 3
Determine the range of f and determine the set of ordered pairs which constitute the graph of f.
Solution: The range will be the set {f(-2), f(-1), f(0), f(1), f(2)}= {1, -2, -3}
The set of ordered pairs is {(-2, 1), (-1, -2), (0, -3), (1, -2), (2, 1)}
Here is a picture of the graph of this function.
72) Consider the function f whose domain is the set {-2, -1, 0, 1, 2}and whose rule is f(x) = |x + 1|
Determine the range of f and determine the set of ordered pairs which constitute the graph of f.
Solution: The range will be the set {f(-2), f(-1), f(0), f(1), f(2)}= {1, 0, 2, 3}
The set of ordered pairs is {(-2, 1), (-1, 0), (0, 1), (1, 2), (2, 3)}
Here is a picture of the graph of this function.
77) Consider the funtion whose rule is f(x) = x2 - x + 1.
Evaluate and simplify 
Solution: Notice that the rule for the function tells us precisely how to calculate the unique range element associated with a domain element. The rule is to subtract the domain element from its square and then add one to that difference.
I this problem, the some of the domain elements are given as algebraic expressions rather than a single number or variable. That does not change the rule of the function.
To calculate f(2), the unique range element associated with 2, we subtract 2 from its square and add one to that difference.
To calculate f(2 + h), the unique range element associated with 2 + h, we subtract 2 + h from its square and add one to that difference.
81) Consider the funtion whose rule is 
.
Evaluate and simplify 
Solution: Notice that the rule for the function tells us the unique range element associated with a domain element is the reciprocal of the square of the domain element
