Overview: 

 

• Features of functions 

• Composite Functions  

• Inverse Functions 

• Sketching Quadratic Functions 

• Sketching Piecewise Functions 

 

 

Features of Functions 

 

What is a function: 

 

• A function is a relation or expression involving one or more variables that gives a results when a value is inputted 

• The notation f(x) is usually used to represent a function 

 

What is the domain of a function: 

 

• The domain is the set of values that can be plugged into a function 

 

What is the range of a function: 

 

• The range is the set of values that can be outputted from a function 

 

 

Composite Functions 

 

 

What is a composite function: 

 

• It is the result of two or more functions being combined into one 

• Consider the question: 

 

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• Here we would plug g(x) into f(x) as we can see that f is on the outside, while g(x) is inside f(x): 

 

 

 

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• Therefore our answer to fg(x) is 8x – 1 

 

 

Practice question:

 

 

 

 

 

 

 

Solution: 

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Inverse Functions 

 

 

What is the inverse function: 

 

• This is a function that is the reverse of another function 

• It is when a function is reflected in the line y = x, causing all the Xs to become Ys and all the Ys to become Xs 

• The inverse function notation is f-1(x) 

 

 

How to find the inverse function: 

 

• Let's consider the function: 

 

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• In order to make the inverse we must let f(x) = y: 

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• Then we need to rearrange the equation to make x the subject: 

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• Then we make the x = f-1(x) and make the y = x: 

 

 

 

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• And this is the answer. 

 

 

 

Practice Questions: 

 

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Solutions: 

 

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Sketching Quadratic Functions 

 

 

How to go about sketching quadratic graphs: 

 

• Let's consider the graph:    x  + 5x + 6 

• Initially we can see that the y intersect of the graph is (0,6), as when x = 0, y = 6  

• Now we need to find the roots by letting y = 0 and solving the quadratic 

• This gets us x = -2 or x = -3, so we cross the x axis at (-2,0) and (-3,0)  

• Finally we need to consider the shape of the graph  

  • As is +x   the curve opens upwards, if it was –x   then the curve would open downwards 

• This leaves us we the graph: 

 

 

 

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Practice Questions: 

 

1.  Sketch the graph y = x  +4x+4 

 

2.  Sketch the graph y = x  +7x+6 

 

 

Solutions: 

 

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Sketching Piecewise Functions: 

 

 

What is a piecewise function: 

 

• This is a function that is made up of multiple different functions over different intervals 

 

How to draw a piecewise function: 

 

• A piecewise function will be made of multiple graph functions 

• In order to draw it you need to draw the individual intervals one at a time 

• At connect them accordingly 

• For example: 

 

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• First we look at the interval 0 ≤ x < 1 and sketch y = x 

 

 

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• Then we look at the interval 1 ≤ x < 2 and sketch y = 1, starting from the end of the previous interval  

 

 

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• And finally we do the same thing for the interval  2 ≤ x ≤ 3 with y = 3 - x