Library of FunctionsYou should be familiar with the shapes of these basic functions. We'll learn them in this section.

Linear Functio

ns

Equations that can be written f(x) = mx + b

The domain of these functions is all real numbers.

slope y-intercept

Constant Functionsf(x) = b, where b is a real number

The domain of these functions is all real numbers.

The range will only be b

f(x) = 3 f(x) = -1 f(x) = 1

Would constant functions be even or odd or neither?

Identity Functionf(x) = x, slope 1, y-intercept = 0

The domain of this function is all real numbers.

The range is also all real numbers

f(x) = x

Would the identity function be even or odd or neither?

If you put any real number in this function, you get the same real number “back”.

Square Functionf(x) = x2

The domain of this function is all real numbers.

The range is all NON-NEGATIVE real numbers

Would the square function be even or odd or neither?

Cube Functionf(x) = x3

The domain of this function is all real numbers.

The range is all real numbers

Would the cube function be even or odd or neither?

Square Root Function

The domain of this function is NON-NEGATIVE real numbers.

The range is NON-NEGATIVE real numbers

Would the square root function be even or odd or neither?

xxf

Reciprocal FunctionThe domain of this function is all NON-ZERO real numbers.

The range is all NON-ZERO real numbers.

Would the reciprocal function be even or odd or neither?

x

xf1

Absolute Value FunctionThe domain of this function is all real numbers.

The range is all NON-NEGATIVE real numbers

Would the absolute value function be even or odd or neither?

xxf

Exponential Function

f(x) = xe

The domain of this function is (-∞, ∞)

The range of this function is {y| y > 0 }

Would the exponential function be even or odd or neither?

Logarithmic Graph

f(x) = ln x

The domain of this function is {x| x > 0}

The range of this function is (-∞,∞)

Would the absolute value function be even or odd or neither?

f(x) = sin x

Sine FunctionThe domain of this function is all real numbers

The range of this function is [ -1, 1]

Would the sine function be even or odd or neither?

f(x) = cos x

Would the cosine function be even or odd or neither?Cosine Function

The domain of the function is all real numbers

The range of this function is [-1, 1]

The Greatest integer Function

The domain of this function is all real numbers

The range of this function is all integers

f(x) = int (x)

Would the greatest integer function be even or odd or neither?

The Logistic Function

f(x) = xe11

The domain of this function is all real numbers

The range of this function is 0 < y <1

Would the logistic function be even or odd or neither?

WISE

FUN

CTIO

NS

These are functions that are defined differently on different parts of the domain.

0,

0,2 xx

xxxf

This means for x’s less than 0, put them in f(x) = -x but for x’s greater than or equal to 0, put them in f(x) = x2

What does the graph of f(x) = -x look like?

Remember y = f(x) so let’s graph y = - x which is a line of slope –1 and y-intercept 0.

Since we are only supposed to graph this for x< 0, we’ll stop the graph at x = 0.

What does the graph of f(x) = x2 look like?

Since we are only supposed to graph this for x 0, we’ll only keep the right half of the graph.

Remember y = f(x) so lets graph y = x2 which is a square function (parabola)

This then is the graph for the piecewise function given above.

0,5

0,3

03,52

xx

x

xx

xf

For x values between –3 and 0 graph the line y = 2x + 5.

Since you know the graph is a piece of a line, you can just plug in each end value to get the endpoints. f(-3) = -1 and f(0) = 5

For x = 0 the function value is supposed to be –3 so plot the point (0, -3)

For x > 0 the function is supposed to be along the line y = - 5x.

Since you know this graph is a piece of a line, you can just plug in 0 to see where to start the line and then count a – 5 slope.

So this the graph of the piecewise function

solid dot for "or equal to"

open dot since not

"or equal to"