Functions. is the INPUT and is called the DOMAIN. is the OUTPUT and is called the RANGE.

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1 Functions Academic Skills Advice Function notation is another way of writing equations. For example: instead of writing y = 7x + 3, we could write f(x) = 7x + 3 (See lesson 2 for more information about function notation). x f(x) is the INPUT and is called the DOMAIN. is the OUTPUT and is called the RANGE. A function should only have one y-value for every x-value. f(x) = x is a function because when you put a value in there is only one possible answer. g(x) = x is not a function as each value you put in has 2 possible answers (e.g. 9 = ±3). Evaluating functions with numbers: We looked at evaluating functions in lesson 2, but here is a recap: If you are asked to evaluate a function you need to put in the x-value (domain) and find the resulting y-value (range). Given the function: f(x) = x + 4 Find f(8) This just means wherever you see an x replace it with an 8. f(x) = x + 4 f(8) = (8) + 4 f(8) = 4 Given the function: g(x) = 4x 2 8x + 2 Find g(3) Now wherever you see an x replace it with a 3. g(x) = 4x 2 8x + 2 g(3) = 4(3) 2 8(3) + 2 g(3) = g(3) = 14 H Jackson 2012 / ACADEMIC SKILLS 1

2 Evaluating functions with Algebra: Whatever you are asked to find the function of just replace the x with it, no matter how complicated. Given the function: h(x) = 3x 7 Find h(x + 2) Wherever you see an x replace it with (x + 2). h(x) = 3x 7 Replace x with (x + 2): h(x + 2) = 3(x + 2) 7 Tidy up: = 3x = 3x 1 Given the function: f(x) = x 2 + 2x 4 Find f(x 3) Wherever you see an x replace it with (x 3). f(x) = x 2 + 2x 4 Replace x with (x 3): f(x 3) = (x 3) 2 + 2(x 3) 4 Multiply all brackets out: = x 2 6x x 6 4 Tidy up: = x 2 4x 1 Given the function: g(x) = 4x + 3 Find g(x 2 x + 2) Wherever you see an x replace it with (x 2 x + 2). g(x) = 4x + 3 Replace x with (x 2 x + 2): g(x 2 x + 2) = 4(x 2 x + 2) + 3 Multiply out bracket: = 4x 2 + 4x Tidy up: = 4x 2 + 4x 5 H Jackson 2012 / ACADEMIC SKILLS 2

3 Composite Functions: You may be asked to combine 2 functions. This can be written as fg(x) or f(g(x)). To work it out you apply g (the inside function) first then evaluate f. If you were asked to do gf(x) you would apply f (the inside function) first then evaluate g. Given the functions: f(x) = x and g(x) = 2x 1 Find: fg(2) gf(4) g 2 (2) fg(x) fg(2) Find g(2): g(2) = 2(2) 1 = 3 Find f(ans): f(3) = = 11 gf(4) Find f(4): f(4) = = 18 Find g(ans): g(18) = 2(18) 1 = 35 g 2 (2) this is the same as doing gg(2) Find g(2): g(2) = 2(2) 1 = 3 Find g(ans): g(3) = 2(3) 1 = 5 fg(x) Find g(x): g(x) = 2x 1 Find f(ans): f(2x 1) = (2x 1) = 4x 2 4x = 4x 2 4x + 3 Inverse Functions: If you have a function f(x) then the notation for the inverse function is f 1 (x). You find an inverse function by undoing the original function. So if the original ended up with add 3 then the inverse would start with subtract 3 etc. This is the method to find the inverse: Replace the function notation with y = Rearrange the function to make x the subject Replace x with y and vice versa. Put the function notation back in. (As we are undoing the original function no matter what f(x) is f 1 (f(x)) = x.) Work through the following examples for more guidance on finding inverse functions. H Jackson 2012 / ACADEMIC SKILLS 3

4 Given the function: f(x) = 7x 4, find f 1 (x) Replace the function notation with y =: y = 7x 4 Rearrange the function to make x the subject: (See Lesson 3 for help with rearranging) Replace x with y and y with x: x = y y = x Put the function notation back in: f 1 (x) = x Given the function: g(x) = x 2 + 3, find g 1 (x) Replace the function notation with y =: Rearrange the function to make x the subject: Replace x with y and y with x: Put the function notation back in: y = x x = y 3 y = x 3 g 1 (x) = x 3 Asymptotes: Some functions have certain values of x which cannot be used. For example: f(x) = 4 x 2 In this function x cannot equal 2 because that would make the bottom of the fraction 0 and you cannot divide by 0. If you were to plot this function on a graph then there would be a line at x = 2 which the graph cannot cross, this line is known as an asymptote. Asymptote H Jackson 2012 / ACADEMIC SKILLS 4

5 Finding the Domain and the Range: The idea of asymptotes can be used to help to find the domain of a function. Domain The domain is all the values that can go into the function. Probably the easiest way to find the domain is to look for any values that can t go in. In the example above the domain would be: all real numbers except 2 (i.e. x R, x 2) The main things to look for are: can t divide by 0 can t square root a negative number x is a member of the real numbers. Find the domain of the function f(x) = 3 x+5 (think of any values that cannot be put into the function) Domain: x R, x 5 (x can be anything except -5) Find the domain of the function g(x) = x 3 Domain: x R, x 3 (x has to be 3 or above so that the number in the square root sign isn t negative) Range The range is all the values that can come out of the function. To find the range just use common sense and think if there is anything that can t come out of the function. Also a function should only have one possible answer so you need to limit the range to validate the function. Find the range of the function f(x) = 3 x+5 (think of any values that cannot come out of the function) Range: f(x) R, f(x) 0 (not possible to divide 3 and end up with 0) Find the range of the function g(x) = x 3 N.b. if the x is on the top then we could get 0 as 0 anything = 0. Range: g(x) 0 (By specifying that the answer has to be greater than 0 we have validated the function as there is now only 1 possible answer for each x value) H Jackson 2012 / ACADEMIC SKILLS 5