Precalculus Chapter 1

The graph of a function

Is a collection of ordered pairs (x, f(x)) such that x is in the domain of f.

Vertical and Horizontal Shifts

Let c be a positive real number. Vertical and horizontal shifts in the graph of y = f(x) are represented as follows 1. Vertical shift c units upward: h(x) = f(x) + c 2. Vertical shift c units downward: h(x) = f(x) - c 3. Horizontal shift c units to the right: h(x) = f(x - c) 4. Horizontal shift c units to the left: h(x) = f(x+c)

Midpoint formula

M=(((x_2+x_1))/2- ((y_2+ y_1))/2)

determine if f(x) = x⁴ has an inverse, if it does, what is it?

No inverse. It fails the horizontal line test.

Write a linear function f such that it has the indicated function values, and sketch the graph of the function f(2/3) = -15/2, f(-4)=-11

Show me the work after you have worked out the problem

Definition of composition of two functions

The composition of the function f with the function g is (f ○ g)(x) = f(g(x)). The domain of f ○ g is the set of all x in the domain of g such that g(x) is in the domain of f.

The graph of an equation

The set of all points that are solution of the equation

Implied domain

The set of all real numbers for which the expression is used to define the function

Zeros of a function

The values for which f(x) = 0

Pythagorean Theorem

a^2+b^2=c^2

determine if x³/8 has an inverse, if it does, what is it?

cube root of 8x

Distance Formula

d= √(〖(x_2-x_1)〗^2+〖(y_2 - y_1)〗^2 )

determine if f(x)=(5 - 3x)/2 has an inverse, if it does what is it?

f⁻¹ (x) = (5 - 2x)/3

determine if 2x has an inverse, if it does what is it?

f⁻¹ (x) = x/2

Rectangular coordinate system

is formed by using real number lines intersecting at right angles

The equation for the slope of a line passing through two points

m= (y_(2 )- y_1)/(x_2-x_1 )

does f(x) = 10 have an inverse, apply the horizontal line test

no

use the geogebra and the horizontal line test to determine if 9 - x² has an inverse

no

Write a linear function f such that it has the indicated function values, and sketch the graph of the function f(-3) = -8, f(1) = 2

show me the work after you have worked out the problem

y-coordinates

the directed distance from the x-axis

x-coordinates

the directed distance from the y-axis

x-axis

the horizontal real number line

quadrants

the plane divided into four equal parts

origin

the point of intersection of the x- and y-axis

ordered pair

the points of correspondence between real numbers on the x- and y-axes.

y-axis

the vertical real number line

Vertical line

x = a

Horizontal line

y = b

Point-slope form of a linear equation

y- y_1=m(x- x_1)

Two-point form

y-y_1= (y_2- y_1)/(x_2- x_1 ) (x- x_1)

The slope-intercept form of a linear equation

y=mx+b

use geogebra to graph (4 - x)/6 and then apply the horizontal line test to determine whether the function is one-to-one and so an inverse function.

yes, it has an inverse

f(x) = x + 2 , g(x) = x - 2 (f + g)(x) = (f*g)(x)= (f - g)(x) = (f/g)(x) = * use geogebra to verify your answers.

(f + g)(x) = 2x (f*g)(x) = (f - g)(x) = (f/g)(x) =

sum

(f + g)(x) = f(x) + g(x)

difference

(f - g)(x) = f(x) - g(x)

Difference quotients

(f(x+h)-f(x))/h

Average rate of change

(f(x_2 )-f(x_1))/(x_2-x_1 )

quotient

(f/g)(x) = f(x)/g(x), g(x) ≠ 0

product

(fg)(x) = f(x) * g(x)

Standard form of an equation of a circle

(x-h)^2+(y-k)^2=r^2

Increasing, decreasing, and constant functions

1. A function is increasing on an interval if, for any x1 and x2 in the interval, x1 < x2 implies f(x1) < f(x2). 2. A function is decreasing on an interval if, for any x1 and x2 in the interval, x1 < x2 implies f(x1) > f(x2) 3. A function is constant on an interval if, for any x1 and x2 in the interval, f(x1) = f(x2).

Tests for even and odd

1. A function y = f(x) is even if for each x in the domain of f, f(-x) = f(x) 2. A function y = f(x) is odd if for each x in the domain of f, f(-x) = -f(x)

Graphical Tests for Symmetry

1. A graph is symmetric with respect to the x-axis if, whenever (x,y) is on the graph, (x,-y) is also on the graph 2. A graph is symmetric with respect to the y-axis if, whenever (x,y) is on the graph, (-x,y) is also on the graph 3. A graph is symmetric with respect to the origin if, whenever (x,y) is on the graph, (-x,-y) is also on the graph.

Parent Functions

1. Constant function 2. Identity function 3. Absolute value 4. Square root function 5. Square function 6. Cubic function 7. Reciprocal function 8. Greatest integer function

Characteristics of a function from set A to Set B

1. Each element in A must be matched with an element in B 2. Some elements in B may not be matched with any element in A 3. Two or more elements in A may be matched with the same element in B 4. An element in A (the domain) cannot be matched with two different elements in B.

Sketching the graph of the equation by plotting points

1. If possible rewrite the equation so that one of the variables is isolated on one side of the equation 2. Make a table of values showing several solution points 3. Plot these points on a rectangular coordinate system 4. Connect the points with a smooth curve or line

Reflections in the Coordinate Axes

1. Reflection in the x-axis: h(x) = -f(x) 2. Reflection in the y-axis: h(x) = f(-x)

Characteristics of the reciprocal function

1. The domain of the function is (-∞,0) U (0,∞) 2. The range of the function is (-∞,0) U (0,∞) 3. The function is odd 4. The graph does not have any intercepts 5. The graph is decreasing on the intervals (-∞,0) and (0,∞) 6. The graph is symmetric with respect to the origin

Characteristics of the square root function

1. The domain of the function is the set of all nonnegative real numbers 2. The range of the function is the set of all nonnegative real numbers. 3. The graph has an intercept at (0,0) 4. The graph in increasing on the interval (0,∞)

Characteristics of piecewise and step functions

1. The domain of the function is the set of all real numbers 2. The range of the function is the set of all integers 3. The graph has a y-intercept at (0,0) and x-intercept in the interval [0,1) 4. The graph is constant between each pair of consecutive integers. 5. The graph jumps vertically one unit at each integer value.

Characteristics of the squaring function

1. The domain of the function is the set of all real numbers 2. The range of the function is the set of all nonnegative real numbers 3. The function is even 4. The graph has an intercept at (0,0) 5. The graph is decreasing on the interval (-∞,0) and increasing on the interval (0,∞) 6. The graph is symmetric with respect to the y-axis 7. The graph has a relative minimum at (0,0)

Characteristics of the cubic function

1. The domain of the function is the set of all real numbers 2. The range of the function is the set of all real numbers 3. The function is odd 4. The graph has an intercept at (0,0) 5. The graph is increasing on the interval (-∞,∞) 6. The graph is symmetric with respect to the origin

Characteristics of Linear Functions

1. The domain of the function is the set of all real numbers 2. The range of the function is the set of all real numbers 3. The graph has an x-intercept of (-b/m,0) and a y-intercept of (0,b) 4. The graph is increasing if m>0, decreasing if m<0, and constant if m=0

Algebraic Tests for Symmetry

1. The graph of an equation is symmetric with respect to the x-axis if replacing y with -y yields an equivalent equation. 2. The graph of an equation is symmetric with respect to the y-axis is replacing x with -x yields an equivalent equation 3. The graph of an equation is symmetric with respect to the origin if replacing x with -x and y with the - y yields an equivalent equation

Finding the intercepts

1. To find the x-intercepts, let y be zero and solve the equation for x 2. To find the y-intercepts, let x be zero and solve the equation for y

Parallel and Perpendicular Lines

1. Two distinct nonvertical lines are parallel if and only if their slopes are equal. That is m1 = m2 2. Two nonvertical lines are perpendicular if and only if their slopes are negative reciprocals of each other. That is, m1 = -1/m2

Finding an inverse function

1. Use the horizontal line test to decide whether f has an inverse function 2. In the equation for f(x), replace f(x) by y. 3. Interchange the roles of x and y, and solve for y. 4. Replace y by f -1 in the new equation 5. Verify that f and f -1 are inverse functions of each other by showing that the domain of f is equal to the range of f -1 , the range of f is equal to the domain of f -1 , and f( f -1(x)) = x and f -1 (f(x)) = x

Horizontal line test for inverse functions

A function f has an inverse function if and only if no horizontal line intersects the graph of f at more than one point

One-to-one functions

A function f is a one-to-one if each value of the dependent variable corresponds to exactly one value of the independent variable. A function f has an inverse function if and only if f is a one-to-one.

Definitions of Relative minimum and relative maximum

A function value f(a) is called a relative minimum of f if there exists an interval (x1,x2) that contains a such that x1 < x < x2 implies f(a) ≤ f(x) A function value f(a) is called a relative maximum of f if there exists an interval (x1, x2) that contains a such that x1 < x < x2 implies f(a) ≥ f(x)

Equation in two variables

A relationship between two quantities

Vertical line test for functions

A set of points in a coordinate plane is the graph of y as a function of x if and only if no vertical line intersects the graph at more than one point.

Piecewise-defined function

An equation defined by two or more equations over a specified domain

Solution or solution point

An ordered pair (a,b) is the solution of an equation if the equation is true

Rigid transformations

Are horizontal shifts, vertical shifts, and reflections

Nonrigid transformations

Are those that cause distortions: vertical stretches, vertical shrinks, horizontal stretches, and horizontal stretches

General Form of a linear equation

Ax + By + C = 0

Domain of an arithmetic combination of functions

Consists of all real numbers that are common to the domains of the functions that were combined

A function

From a set A to a set B is a relation that assigns to each element x in the set A exactly one element y in the set B. The set A is the domain, and the set B is the range.