Inverse Functions Reversing the process

Consider the function f(x) = 2x + 1 Set A Set B –2 5 –3 1 11 The function which changes the numbers in Set B back into the numbers in set A is called an INVERSE function Make up and example of a relation then define the domain and range of the relation The opposite of 2x + 1 is take 1 and divide by 2

Consider the function f(x) = 2x + 1 Set A Set B –2 5 –3 1 11 Make up and example of a relation then define the domain and range of the relation g(x) is the inverse of f(x), usually written as f -1(x)

g(x) is the inverse of f(x), usually written as f -1(x) You have seen this notation before when using your calculator sin-1, cos-1, tan-1 are the inverse functions of sin, cos and tan Make up and example of a relation then define the domain and range of the relation

Graphically the inverse function is the reflection of the given function in the line y = x [ f(x) = 2x + 1] Make up and example of a relation then define the domain and range of the relation

Exponential & Log Functions

Exponential (to the power of) Graphs A function in the form f(x) = ax where a > 0, a ≠ 1 is called an exponential function to base a . Consider f(x) = 2x x -3 -2 -1 0 1 2 3 f(x) 1/8 ¼ ½ 1 2 4 8 Remember a0 = 1 and a1 = a

Exponential (to the power of) Graphs y = 2x A growth function (1 , 6) (2 , 4) y = 6x (1 , 2)

The graph of y = ax always passes through (0 , 1) & (1 , a)

y = 2x y = x y = log2x

To obtain y from x we must ask the question Log Graphs ie x 1/8 ¼ ½ 1 2 4 8 y -3 -2 -1 0 1 2 3 To obtain y from x we must ask the question “What power of 2 gives us…?” This is not practical to write in a formula so we say “the logarithm to base 2 of x” y = log2x or “log to the base 2 of x”

y = log2x (2 , 1) NB: x > 0 Major Points (i) y = log2x passes through the points (1,0) & (2,1) . As x  ∞, y   but at a very slow rate and as x  0 y  – ∞

The graph of y = logax always passes through (1,0) & (a,1)

→ log 2 8 = 3 → log 10 10000 = 4 → log 6 36 = 2 → log a c = b You should note carefully the connection between Exponentials and Logs. Exponential Log → log 2 8 = 3 23 = 8 → log 10 10000 = 4 104 = 10000 → log 6 36 = 2 62 = 36 → log a c = b ab = c

(1 , a) (a , 1) (0 , 1) (1 , 0)

Determine the equation of the graph. The diagram shows the graph of y = f (x) where f is a logarithmic function. (6 , 1) 1 (4 , 0) +3 Determine the equation of the graph. y = loga (x – 3) When x = 6, y = 1 1 = loga 3  a = 3 y = log3 (x – 3)

The diagram shows the graph of y = a log2(x + c) (6 , 9) The diagram shows the graph of y = a log2(x + c) –1 1 –2 Determine the values of a and c. y = a log2 (x + 2)  c = 3 When x = 6, y = 9 9 = a log2 8  9 = a × 3  a = 3

The diagram shows part of the graph whose equation is of the form y = 2mx. What is the value of m? (3 , 54) When x = 3, y = 54 54 = 2 × m3 m3 = 27 m = 3

Now try Exercise 7 page 42 1, 2, 3a), 4a) and 8