# Functions

Contents

Functions are mathematical entities that assign unique outputs to given inputs. Sounds simple? Think again!
In this topic you will evaluate, graph, analyze, and create various types of functions.

## Introduction to functions

Make a first introduction to functions.

What is a function?

Functions assign a single unique output for each of their inputs. In this video, Sal gives many examples of various kinds of functions.

## Evaluating functions

Learn how to find the value of a function for a given input value.

Evaluating a function: formula

Sal evaluates f(x)=49-x^2 at x=5.

Evaluate functions

Evaluate functions for specific inputs given the formula of the function.

Evaluate a function: graph

Sal evaluates a function at x=-1 using the graph of that function.

Evaluate functions from their graph

Evaluate functions for specific inputs given the graph of the function.

How to evaluate an expression that contains function notation (example)

Sal evaluates -2*f(-6)+g(1) given the graphs of f and g.

Evaluate function expressions

Evaluate expressions that contain function notation, given the functions' graphs.

## Inputs and outputs of a function

Extend your understanding of the relationship between the inputs of a function and the outputs of that function.

How to match function input to output given the formula (example)

Sal finds the input value for which f(t)=13, given that f(t)=-2t+5.

Match function inputs to outputs from a formula

Find all the inputs that correspond to a given function output, using the function's formula.

How to match function input to output given the graph (example)

Sal finds the input value for which g(x)=-2 given the graph of g.

How to find two function inputs with the same output given graph (example)

Sal finds the input value other than -5 for which f(x)=f(-5), given the graph of f.

Match function inputs to outputs from a graph

Find all the inputs that correspond to a given function output, using the function's graph.

## Functions and equations

Understand the subtle differences and similarities between functions and equations. In this exercise, we will see how an equation can be turned into a function.

Equations vs. functions

Jesse Roe and Sal talk about the difference between equations and functions

How to create a function from an equation (example)

Sal find the formula of a function that, given an input value "b," outputs a value "a" that satisfies the equation 4a+7b=-52.

Function rules from equations

Create functions that match one variable to the other in a two-variable equation.

## Interpreting function notation

Solve some word problems by interpreting expressions of modeling functions.

How to interpret an expression with function notation (example with a function that models a bank account)

Sal interprets the expression M(30)-M(0)=100 where M models an account balance over time.

How to interpret an expression with function notation (example with a function that models the number of people at the beach)

Sal interprets the expression P(5)<P(9) where P models the number of people at the beach over time.

Function notation interpretation word problems

Interpret expressions with function notation in terms of the context that the function models.

## Introduction to the domain and range of a function

Learn what the domain and the range of a function are. Practice finding the domain and the range of a function given its graph.

Intervals and interval notation

Intervals describe specific sets of numbers and are very useful when discussing domain and range. In this video, Sal introduces the different types of intervals and their notations.

What is the domain of a function?

Sal introduces the concept of "domain" of a function and gives various examples for functions and their domains.

What is the range of a function?

Sal introduces the concept of "range" of a function and gives examples for functions and their ranges.

How to find the domain and the range of a function given its graph (example)

Sal solves multiple examples where the graph of a function is given and the question asks for the domain or the range of that function.

Domain and range from graph

Given the graph of a function, determine its domain or range.

## Determining the domain of a function

Determine the domains of functions according to various considerations.

Domain of a radical functions

Sal finds the domain of f(x)=√(2x-8).

How to determine the domain of algebraic functions (examples)

Sal gives many examples where he determines the domains of functions according to mathematical limitations.

Determine the domain of functions

Determine the domain of a function according to the algebraic limitations of that function.

How to determine the domain of a modeling function (example with a function that is defined over the real numbers)

Sal determines the domain of a function that models the height of a plant over time.

How to determine the domain of a modeling function (example with a function that is defined over the positive integers)

Sal determines the domain of a function that models the price of candy bars.

How to determine the domain of a modeling function (example with a function that is defined over negative and positive integers)

Sal determines the domain of a function that models going up and down a ladder.

Function domain word problems

Use information about a situation to figure out the domain of the function which models it.

## Recognizing functions

Recognizing functions

Checking if a set of points represents a function

Sal checks whether a given set of points can represent a function. For the set to represent a function, each domain element must have one corresponding range element at most.

Does a vertical line represent a function?

Sal explains why a vertical line *doesn't* represent a function.

Recognize functions from graphs

Determine whether a given graph represents a function.

Checking if two quantities represents a function: heights

Sal checks whether a table of people and their heights can represent a function that assigns a height to a name.

Recognize functions from tables

Determine whether a table of values of a relationship represents a function.

Is this a function? (verbal description of equation)

Sal checks whether y can be described as a function of x if y is always three more than twice x.

Checking if two quantities represents a function: price & cost

Sal checks whether a description of the price of an order can be represented as a function of the shipping cost.

## Piecewise functions

Introduction to piecewise defined functions

Introduction to piecewise functions

Sal explains what a piecewise function is, and finds the formula of a piecewise function given its graph.

How to evaluate a piecewise function from its formula or graph (example)

Sal evaluates piecewise functions at given inputs, both from a formula and from a graph.

Evaluate piecewise functions from the formula

Given the formula of a piecewise function, evaluate it for a specific input.

Evaluate step functions from their graph

Given the graph of a step function, find the function's outputs for given specific inputs.

How to graph a piecewise function (example)

Sal graphs a piecewise function given its formula. In this case, the function consists of three separate lines.

Graphs of piecewise linear functions

Draw the graph of a piecewise function or determine its formula from its graph.

How to find the domain and range of a piecewise function (example with a step function)

Sal finds the domain and range of a piecewise function that is constant in each segment. Such functions are called "step functions."

How to find the domain and range of a piecewise function (example with a piecewise-linear function)

Sal finds the domain and range of a piecewise function where each segment is linear.

Domain and range of piecewise functions

Find the domain and range of a piecewise function given its formula.

## Maximum and minimum points

Learn about maximum and minimum points of functions.

Introduction to minimum and maximum points

Sal explains all about minimum and maximum points, both absolute and relative.

How to recognize relative and absolute maxima and minima (example)

Sal gives multiple examples where he identifies relative maximum points, relative minimum points, absolute maximum points, and absolute minimum points.

Recognize relative maxima and minima

Given the graph of a function, find all of its relative maximum and minimum points.

Recognize absolute maxima and minima

Given the graph of a function, find its absolute maximum and minimum points.

## Intervals where a function is positive, negative, increasing, or decreasing

Learn about features of functions that have distinct graphical representations: intervals where the function is always positive or always negative, and intervals where the function is always increasing or always decreasing.

Introduction to increasing, decreasing, positive or negative intervals

Sal discusses there Intervals where function is increasing, decreasing, postive or negative and their graphical representation.

How to identify positive and negative intervals (example)

Sal gives several example of graphs where he highlights positive or negative intervals of the function.

Positive and negative intervals

Highlight intervals on the domain of a function where it's entirely positive or entirely negative.

Increasing and decreasing intervals

Highlight intervals on the domain of a function where it's only increasing or only decreasing.

## Interpreting features of graphs

Interpret the graphs of functions in terms of the contexts that are modeled by the functions.

How to match the features of a modeling function to their real-world meaning (example)

Sal considers the graph of a function that models temperature over time, and matches features of that graph to their real-world meaning.

How to interpret the y-intercept of a graph in context (example)

Sal discusses the meaning of the y-intercept of the graph of a function that models the height of a ball.

Graph interpretation word problems

Match features of graphs of modeling functions to their real-world meaning.

## Average rate of change

Learn what's the average rate of change of a function and how to find it over given intervals.

Introduction to average rate of change

Sal explains what's the average rate of change of a function over an interval.

How to find the average rate of change from a graph (example)

Sal finds the interval in a function's graph where the function has an average rate of change of -4.

How to find the average rate of change from a table (example)

Sal finds the average rate of change of a function over the interval -5<x<-2, given a table of values of the function.

How to find the average rate of change from a formula (example)

Given the formula of a function, Sal finds the interval where the function has an average rate of change of 1/2.

Finding average rate of change

Find the average rate of change of a function over a given interval.

## Average rate of change word problems

Solve word problems that concern the average rate of change of a functional relationship.

How to find the average rate of change of a modeling function from a table (example)

Sal compares the average rates of a change of a function that models temperature for different time intervals.

How to estimate the average rate of change of a modeling function from a graph (example)

Given the graph of a skydiver's altitude, Sal finds the average rate of change in altitude over a given time interval.

How to represent the average rate of change of a modeling function as an expression (example)

Sal finds the algebraic expression that represents the average rate of change of water in a drained bucket over a given time interval.

Average rate of change word problems

Find and represent the average rate of change of a real-world relationship.