MAT01A1: Inverse Functions Dr Craig 27 February 2018
Introduction Who: Dr Craig What: Lecturer & course coordinator for MAT01A1 Where: C-Ring 508 acraig@uj.ac.za Web: http://andrewcraigmaths.wordpress.com
Collecting assignments/class tests Scripts from Assignment 2 can be collected at the collection facility (C-Ring 511). Opening hours: Mon Fri: 09h00 12h00 and 13h00 15h00 Learn from these assessments! Maths Learning Centre in C-Ring 512: 10h30 15h00 Mondays 08h00 15h30 Tuesday to Thursday 08h00 12h55 Fridays
Important information Lecture times: Tuesday 08h50 10h25 Wednesdays 17h10 18h45 Lecture venues: C-LES 102, C-LES 103 Tutorials: Tuesday afternoons 13h50 15h25: D-LES 104 or D-LES 106 OR 15h30 17h05: C-LES 203 ONLY
Lecturers Consultation Hours Monday: 12h00 13h30 Ms Richardson (C-503) Wednesday: 15h00 16h00 Ms Richardson (C-503) Thursday: 11h20 12h55 Dr Craig (C-508) Friday: 11h20 12h55 Dr Craig (C-508)
Warm-up (from last time): rewrite and simplify the following expressions: a b 5 a, b > 0 ab x 4x x 5x+1 x x 5 Exponential graph example: Sketch y = 4 5 x. (Hint: rewrite as y = 5 x + 4.)
Why do we want inverse functions? Many functions give us the quantity of y based on the value of x. What if we want to find out what the x value is when a certain quantity of y is reached? The independent variable (input variable) of a function is often time (t) and it is important to be able to know how long it will take for a certain output value to be reached.
The independent variable (input variable) of a function is often time (t) and it is important to be able to know how long it will take for a certain output value to be reached. Example: exponential growth and decay y = f(t) = ae kt t = 1 k ln ( y a )
One-to-one functions In the diagrams above, f : A B is a one-to-one function, but g : A B is not.
A function f is called a one-to-one function if f(a) = f(b) = a = b This can also be stated in the contrapositive: a b = f(a) f(b) Example: show that f(x) = x 2 x + 2 is 1-1.
Horizontal line test A function is one-to-one if and only if no horizontal line intersects its graph more than once. Examples: is f(x) = x 3 one-to-one? is g(x) = x 2 one-to-one?
Definition: inverse functions Let f be a one-to-one function with domain A and range B. Then its inverse function f 1 has domain B and range A and is defined by f 1 (y) = x f(x) = y for any y B.
Very important: f 1 (x) 1 f(x) If we want to denote the reciprocal of f(x) we would write 1 f(x) = (f(x)) 1
How to find the inverse of f(x) 1. Write y = f(x). 2. Solve this equation for x in terms of y. 3. Express f 1 as a function of x by swapping x and y. Examples: f(x) = x + 3 f(x) = x 2 x + 2
f(x) = x 2 x + 2
f(x) = x 2 x + 2 f 1 (x) = 2x 2 x 1
f(x) = x 2 x + 2 f 1 (x) = 2x 2 x 1
Sketching inverse functions The graph of f 1 is obtained by reflecting the graph of f about the line y = x.
Log functions as inverses If a > 0 and a 1 then the exponential function f(x) = a x is either increasing (if a > 1) or decreasing (a < 1). Such an exponential function will never have two x values x 1 and x 2 such that a x 1 = a x 2. Therefore it is 1-1 and has an inverse function given by f 1 (x) = log a x If a = e then we write f 1 (x) = ln(x).
Log laws If x, y > 0 then (a) log a (xy) ) = log a x + log a y (b) log a y = log a x log a y (c) log a (x r ) = r log a x (for x R) Examples: Solve for x log 10 5 + log 10 (5x + 1) = log 10 (x + 5) + 1 log 42 log 6 Compute log 6 216 + log 49
Change of base law For any positive number a (a 1): log a x = ln x ln a Hyperlink to Khan Academy exercise set: Change of base
Khan Academy exercises Click on the hyperlinks below to practise: Logs 1 Logs 2 Logs 3 Graphs of exponential and log functions
Inverse trig functions The function f(x) = sin x is not one-to-one when x R. However, we can consider a piece of it which is one-to-one. We restrict the domain so that x [ π 2, π 2 ]. (We could take other 1-1 pieces of the function, but this interval is considered to be the standard option.)
Important point about notation: sin 1 (x) 1 sin x 1 sin x = (sin x) 1 sin 1 (x) = arcsin x
cos x and arccos x: again, we must choose a one-to-one piece of the curve y = cos x. It is standard to use x [0, π]. With the restricted domain we have: ran(cos x) = dom(cos 1 (x)) & dom(cos x) = ran(cos 1 (x))
y = tan 1 x = arctan x NB: tan 1 (x) cot(x).
Some inverse trig examples Calculate ( the following ) arcsin 3 2 cos(arcsin t) cos(arctan 2 3 ) Tough one: sec(arcsin(2x 1))
A nice exercise: Q17 from the textbook asks: If g(x) = 3 + x + e x, find g 1 (4). Recall that x = g 1 (y) y = g(x) This question is therefore asking: what is the input value into g(x) that will give an output value of 4? Hint: you do not need to calculate g 1 (x) to solve this problem.