FINDING THE ZEROS OF A FUNCTION
College Algebra Exercises Section 2.3
A zero of a function f is a domain element
x for which f(x) = 0.
Therefore when we want to find the zeros of any function, we want to
find all domain elements whose corresponding range element is 0.
That is, we want to find all domain elements x for which f(x) =0
15) Find the zeros of the function f whose rule is
f(x)
= 2x2 – 7x -30
Solution: Recall the definition of a
zero
of a
function
to
realize that to
find the zeros
of this function f we must solve the equation
resulting from f(x) = 0.
So we must solve
0
= f(x) = 2x2 – 7x –30
The solutions are x = 6 and x =
and therefore the zeros of the given function f are 6 and
Some Observations: To
verify that 6 and 
are zeros of the function, simply compute f(6) and f( 
) according to the rule f(x) = 2x2 – 7x –30
and observe that f(6) = 0 and f( 
) = 0.
Notice that we now know ( 
, 0) and (6, 0) are on the
graph
of the function f.
In fact ( 
, 0) and (6, 0) are the x-intercepts of the graph of the function f.
Parenthetically: As a
form of review we present the mechanics of solving the equation
0 = 2x2 – 7x –30
(2x + 5)(x – 6) = 0
2x + 5 = 0 or x – 6 = 0
x = 
or x
= 6
16) Find the
zeros of the
function
f whose rule is
f(x)
= 3x2 + 22x -16
Solution: Recall the definition of a zero of a function to
realize that to
find the zeros
of this function f we must solve the equation
resulting from f(x) = 0.
So we must solve
0
= f(x) = 3x2 + 22x -16
The solutions are x = -8 and x =
and therefore the
zeros of the given function f are -8 and 
Some Observations: To
verify that -8 and 
are zeros of the function, simply compute f(-8) and f( 
) according to the rule f(x) = 3x2 + 22x –16 and
observe that f(-8) = 0 and f( 
) = 0.
Notice that we now know ( 
, 0) and (-8, 0) are on the
graph
of the function f.
In fact ( 
, 0) and (-8, 0) are the x-intercepts of the graph of the function
f.
Parenthetically: As a
form of review we present the mechanics of solving the equation
0 = 3x2 + 22x –16
(3x - 2)(x + 8) = 0
3x - 2 = 0 or x + 8 = 0
x = 
or x
= -8
17) Find the
zeros of the
function
f whose rule is
Solution: Recall the definition of a zero of a function to
realize that to
find the zeros
of this function f we must solve the equation
resulting from f(x) = 0.
So we must solve
The solution is x = 0 and therefore the zero of the given function f is
0
Some Observations: To
verify that 0 is the zero of the function, simply compute f(0) and observe that
f(0) = 0
Notice that we now know (0, 0) is on the
graph
of the function f.
In fact (0, 0) is the x-intercept of the graph of the function f.
Parenthetically: As a
form of review we present the mechanics of solving the equation
if and only if the numerator is 0
20) Find the
zeros of the
function
f whose rule is
f(x)
= x3 – 4x2 – 9x + 36
Solution: Recall the definition of a zero of a function to
realize that to
find the zeros
of this function f we must solve the equation
resulting from f(x) = 0.
So we must solve
0
= f(x) = x3 – 4x2 – 9x + 36
The solutions are x = 4, x =
3, and x = -3 and therefore the zeros of the given function f are 4, 3, and -3
Some Observations: To
verify that 4 is the zero of the function, simply compute f(4) and observe that
f(4) = 0
To
verify that 3 is the zero of the function, simply compute f(3) and observe that
f(3) = 0
To
verify that -3 is the zero of the function, simply compute f(-3) and observe that
f(-3) = 0
Notice that we now know (4, 0), (3, 0), and (-3, 0) are on the
graph
of the
function f.
In fact (4, 0), (3, 0), and (-3, 0) are the x-intercepts of the graph of
the function f.
Parenthetically: As a
form of review we present the mechanics of solving the equation
x3 – 4x2 – 9x + 36 = 0
(x3 – 4x2) – (9x – 36) = 0
x2(x – 4) – 9(x – 4) = 0
(x – 4)(x2 – 9) = 0
(x – 4)(x – 3)(x + 3) = 0
x – 4 = 0 or (x
– 3) = 0 or
(x + 3) = 0
x = 4 or x = 3 or
x = -3
24) Find the
zeros
of the
function
f whose rule is
Solution: Recall the definition of a zero of a function to
realize that to
find the zeros
of this function f we must solve the equation
resulting from f(x) = 0.
So we must solve
The solution is x = 
and therefore the zero of the given function f is 
Some Observations: To
verify that 
is the zero of the function,
simply compute f() and observe that f() = 0
Notice that we now know ( 
, 0) is on the
graph
of the function f.
In fact ( 
, 0) is the x-intercept of the graph of the function f.
Parenthetically: As a
form of review we present the mechanics of solving the equation
if and only if 3x +
2 = 0 which implies x =
is the solution of the
equation