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Harder one:
A function is defined by : f(x) = (x+3)(x-5)
Find the inverse and state the domain and range of f(x) and f-1(x)
A function is defined by : f(x) = (x-16)(x-4)
Find the inverse and state the domain and range of f(x) and f-1(x)
Homework:
Functions on moodle with mark scheme
Inverse functions
Recap of inverse of a function.Inverse functions with ex and ln
xHarder inverse functions such
as quadratics and algebraic fractions
A neat little trick…
► As always in maths, there is a trick to this…
1. Write function as a rule in terms of y and x.
2. Swap ‘x’ and ‘y’3. Rearrange to get in terms of y.4. Write as f-1 (x) =
A neat little trick…
32 xxf
32 xy
32 yx
62 yx
yx 26
2
6x
y
Find the inverse of
2
61 x
xf
Inverse functions
Inverse functions only exist for one-one functions.
Things to note..
The domain of f-1 is the range of f and the range of f-1 is the domain of f.
The graph of an inverse function can be found by reflecting a function in the line y=x
Have a go:
Worksheet C
Questions 1-7
ppt: 10 questions
-5
-4
-3
-2
-1
0
1
2
3
4
5
-3 -2 -1 0 1 2 3
y=ln(x) is a reflection of y = ex in the line y = x
y = ex
y = x
y = ln (x)
y = ex, y = x and y = ln x
y=ln(x) and y = ex are inverse functions
Inverse functions with ex
e.g. f(x) = ex -2 x = ey -2
x +2 = ey
ln(x +2) = ln ey
ln(x +2) = y
The inverse of f(x) is …
f-1(x) = ln(x +2)
Domain is x > -2 -5
-4
-3
-2
-1
0
1
2
3
4
5
-3 -2 -1 0 1 2 3
y = ex - 2
y = ln (x+2)
“The graph of an inverse function can be found by reflecting a function in the line y=x”
Inverse functions with ex
e.g. f(x) = e2x-1 + 6 x = e2y-1 + 6
x - 6 = e2y-1
ln(x - 6) = ln e2y-1
ln(x - 6) = 2y - 1
The inverse of f(x) is …
f-1(x) = ½(ln(x-6) + 1)
Domain ?
ln(x - 6) +1 = 2y
½(ln(x - 6) +1) = y Domain is x > 6
Cannot have ln of numbers less than 0
Inverse functions with ln x
e.g. f(x) = ln(2x) + 6 x = ln(2y) + 6
x - 6 = ln (2y)
ex-6 = eln 2y
ex-6 = 2y
The inverse of f(x) is …
f-1(x) = ½ ex-6
Domain ? ½ ex-6 = y
Have a Go
Domino Trailor
Worksheet C
Questions 8,9,11
Extension: Exam Questionsppt: 10 questions
Plenary
Plenary
f(x) = x . x+1
Show that f-1(x) = 1 - 1 1-x
Extension: