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§ 0.2
Some Important Functions
Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 1 of 78
Section Outline

Linear Equations

Applications of Linear Functions

Piece-Wise Functions

Quadratic Functions

Polynomial Functions

Rational Functions

Power Functions

Absolute Value Function
Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 2 of 78
Linear Equations
Equation
Example
y = mx + b
(This is a linear function)
x=a
(This is not the graph of a
function)
Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 3 of 78
Linear Equations
CONTINUED
Equation
Example
y=b
Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 4 of 78
Applications of Linear Functions
EXAMPLE
(Enzyme Kinetics) In biochemistry, such as in the study of enzyme kinetics, one encounters a linear
function of the form f x  K / V x  1/ V , where K and V are constants.
(a) If f (x) = 0.2x + 50, find K and V so that f (x) may be written in the form, f x  K / V x  1/ V.
(b) Find the x-intercept and y-intercept of the line f x  K / V x  1/ V in terms of K and V.
SOLUTION
(a) Since the number 50 in the equation f (x) = 0.2x + 50 is in place of the term 1/V (from the
original function), we know the following.
50 = 1/V
50V = 1
V = 0.02
Explained above.
Multiply both sides by V.
Divide both sides by 50.
Now that we know what V is, we can determine K. Since the number 0.2 in the equation
f (x) = 0.2x + 50 is in place of K/V (from the original function), we know the following.
Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 5 of 78
Applications of Linear Functions
CONTINUED
0.2 = K/V
0.2V = K
0.2(0.02) = K
0.004 = K
Explained above.
Multiply both sides by V.
Replace V with 0.02.
Multiply.
Therefore, in the equation f (x) = 0.2x + 50, K = 0.004 and V = 0.02.
(b) To find the x-intercept of the original function, replace f (x) with 0.
f x  K / V x  1/ V
0  K / V x  1/ V
1/ V  K / V x
This is the original function.
Replace f (x) with 0.
Solve for x by first subtracting 1/V from
both sides.
Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 6 of 78
Applications of Linear Functions
CONTINUED
 1/ V V / K   x
1 / K  x
Multiply both sides by V/K.
Simplify.
Therefore, the x-intercept is -1/K. To find the y-intercept of the original function, we
recognize that this equation is in the form y = mx + b. Therefore we know that 1/V is the
y-intercept.
Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 7 of 78
Piece-Wise Functions
EXAMPLE
1  x for x  3
.
2
for
x

3

Sketch the graph of the following function f x   
SOLUTION
We graph the function f (x) = 1 + x only for those values of x that are less than or equal to 3.
6
4
2
0
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
-2
-4
-6
Notice that for all values of x greater than 3, there is no line.
Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 8 of 78
Piece-Wise Functions
CONTINUED
Now we graph the function f (x) = 4 only for those values of x that are greater than 3.
6
5
4
3
2
1
0
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
Notice that for all values of x less than or equal to 3, there is no line.
Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 9 of 78
Piece-Wise Functions
CONTINUED
Now we graph both functions on the same set of axes.
-6
-5
-4
-3
-2
6
5
4
3
2
1
0
-1 -1 0
-2
-3
-4
-5
-6
1
2
3
4
5
6
Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 10 of 78
Quadratic Functions
Definition
Example
Quadratic Function:
A function of the
form
f x   ax 2  bx  c
where a, b, and c are
constants and a  0.
Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 11 of 78
Polynomial Functions
Definition
Example
Polynomial Function: A
function of the form
f x   an x n  an 1 x n 1    a0
f x   17 x3  x 2  5
where n is a nonnegative
integer and a0, a1, ...an are
given numbers.
Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 12 of 78
Rational Functions
Definition
Rational Function: A
function expressed as the
quotient of two
polynomials.
Example
3x  x 4
g x   2
5x  x  1
Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 13 of 78
Power Functions
Definition
Example
Power Function: A
function of the form
f x   x 5.2
f x   x r .
Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 14 of 78
Absolute Value Function
Definition
Example
Absolute Value Function:
The function defined for
all numbers x by
f x   x
f x   x ,
such that |x| is understood
to be x if x is positive and
–x if x is negative
f 1 2  1 2  1 2
Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 15 of 78