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Review of Functions

• These are review topics from algebra and pre-calculus – it would be good to just briefly “brush up” on these things!
• As you go through calculus, it will be important to use the correct terminology for the various terms associated with functions – clear mathematical communication is important!

Review of Functions

Simplify tanx cscx
• y = tanx cscx
• = [sinx/cosx] [1/sinx]
= [1/cosx] = secx
Simplify cscx cosx
• y = cscx cosx
• = [1/sinx] cosx
• = [cosx/sinx]
=cotx
Simplify [4x sinx/(x2 cos3x)]
• y = [4x sinx/(x2 cos3x)]
• = [4x/(x2)] [sinx/cos3x]
and cannot be combined because their arguments differ. The arguments must be the same to rewrite trig functions. 0Find the domain and range of
• sin and cos are periodic functions, meaning the outputs will repeat once the input is shifted over a full period. The unit circle is a good way to see this. No matter what real numbers we put into the function, the same pattern repeats. The output never exceeds 1 or goes below -1
Domain (), Range [-1, 1] 0Find the domain and range of
This is similar to the previous problem, but it's a bit trickier because it introduces zero into the denominator. The function is said to be undefined where the input results in a zero in the denominator. For this function, we won't have any problems when the denominator is not zero. So the domain involves any x when is not zero. is zero at any multiple of , including zero. Now, we're dividing by really tiny numbers (approaching zero). An integer divided by a very small number, is a very large number. Domain is all real numbers, but cannot equal any multiple of . where n is any integer. Range 0Graph
• This is very similar to the regular cos graph, except that its period is changed, or quished." Its usual period of is now .
0Graph
This is similar to the regular sin graph, but it is shifted UP by one unit. 0Graph
• is very different from . Radians are used unless otherwise specified. By convention, the variable typically implies use of degrees).
• has a period of , meaning that going units in x in either direction will return the same result
• etc...
• so this is the same graph as
0Graph
• The number out front here determines the amplitude. In this case, it's 3. So from minimum to maximum, this graph will cover 6 units total. Multiplying the x here by 2 changes the period from to .
0Graph
• The positive 4 there moves the graph over 4 units to the left. The -1 in that location means the graph is shifted down 1. This is the same graph as the previous problem, but shifted down 1 and 4 to the left.
0Simplify
• This is equivalent to , which would give an exponent of , or
0Rewrite
0Rewrite
0Combine the exponents of
0Combine the exponents of
0Simplify
0Simplify
0Solve for x,
0Solve for x,
0Prove
• ADVANCED NOTE: This is the exponential expression of the trig identity
0Given , find
0Solve for x,
0Simplify
0Solve for x,
x = 1 0Solve for y,
0Solve for x,
0Solve for x,
0Simplify
Here, the base is irrelevant. 0Solve for ,
• The second solution, -9, would be outside of the domain of . For , x must be greater than 0. So
0Prove
• Let and
0Find the roots of
0Find the roots of
Two identical roots, 0Find the asymptote(s) of
We already found the roots on the previous problems. The asymptote is where the function is undefined. Here, it's when the denominator is equal to zero. We know that the denominator is equal to zero when . The function goes to positive infinity when approaching the root () from the positive direction. The function also goes to positive infinity when approaching the root from the negative direction (The degree of the polynomial in the denominator is even). 0Graph
• We already determined the asymptote behavior, what about the edge behavior? As x gets very large and positve, y approaches zero. As x gets very large and negative, y also approaches zero from the positive side.
0Find the asymptote(s) of
The function is undefined at x = 0. The function goes to positive infinity when approaching zero from the positive direction. The function goes to negative infinity when approaching zero from the negative direction (Remember, the polynomial in the denominator is of odd degree). 0Find the asymptote(s) of
• . The function is undefined when . when
There are an infinite number of asymptotes for and those asymptotes are at odd integer multiples of 0Graph
• This is similar to the graph of , but it is shifted 5 to the left.
0Graph
• Even degree polynomial in the denominator. As x gets very large in either direction, y approaches zero. There are two asymptotes here, -2 and 2. That gives us 4 different cases for asymptotic behavior. Approaching -2 from the left, Approaching -2 from the right, approaching 2 from the left, and approaching 2 from the right. Plugging in values to test for these cases.
0Graph
• Asymptotes are at same locations as previous problem, so we can find the asymptote behavior in a similar fashion
0Graph
• Remember the properties of . Domain , , and is an increasing function. As x gets larger, f(x) approaches zero and f(x) has an asymptote at x = 1.

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Review of Functions

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

• Intro 0:00
• What is a Function 0:10
• Domain and Range
• Vertical Line Test
• Example: Vertical Line Test
• Function Examples 1:57
• Example: Squared
• Example: Natural Log
• Example: Exponential
• Example: Not Function
• Odd and Even Functions 4:39
• Example: Even Function
• Example: Odd Function
• Odd and Even Examples 6:48
• Odd Function
• Even Function
• Increasing and Decreasing Functions 10:15
• Example: Increasing
• Example: Decreasing
• Increasing and Decreasing Examples 11:41
• Example: Increasing
• Example: Decreasing
• Types of Functions 13:32
• Polynomials
• Powers
• Trigonometric
• Rational
• Exponential
• Logarithmic
• Lecture Example 1 15:55
• Lecture Example 2 17:51