Inverse of a function
Clement Radcliffe, Contributor

We began the review of functions and relations last week. In today's lesson, we will share the solution to last week's homework.

• Given that f : x --> 3x - 2

g : x --> 2x + 5

Evaluate:

(i) g(-6)

(ii) fg (3)

Solution:

(i) Since g : x --> 2x + 5, g(x) = 2x + 5

... g(-6) = (2 x -6) + 5

= -12 + 5 = -7.

... g(-6) = -7

(ii) Remember now that for fg(x), g(x) replaces x in f(x)

As g(x) = 2x + 5 and f(x) = 3x - 2

... fg(x) = f (2x + 5)

... fg(x) = 3 (2x + 5) - 2

= 6x + 15 - 2 = 6x + 13

... fg(3) = (6 x 3) + 13 = 18 +13

... fg(3) = 31

• If f(x) = 2x - 1 and g(x) = 1/2 (x + 2), calculate

(i) f(3)

(ii) gf(3).

Solution:

(i) As f(x) = 2x - 1, then f(3) = (3 x 2) -1 = 6 - 1 = 5.

... f(3) = 5

(ii) Since f(3) = 5, then gf(3) = g(5) .

Since g(x) = (x + 2)

... g(5) = 1/2 (5 + 2) = 7/2

... gf(3) = 7/2.

Now that we have gone through the homework, our lesson will continue.

Inverse of a function

If f is the function defined as y = ax + b, then f ^-1, the inverse function, expresses the variable x in terms of y.

Example: y = ax + b

...ax = y - b

...x = (y - b)/a ( x is expressed as a function of y)

Interchange x for y. (This is necessary as y is always expressed as a function of x)

... y = (x - b)/a

... f^-1(x) = (x - b)/a or ... f^-1 = (x - b)/a

that is, the inverse of function f, (f ^-1), is (x - b)/a

Please note that this method should always end with the statement:

f ^-1 (x) = (x - b)/a and NEVER y = (x - b)/a.

Given the function y = ax + b, some students express f ^-1(x) as 1/ax + b by assuming that -1 is the power of f as in indices. I am sure you will never make this error.

Example 1: Given that f(x) = 1/2(x + 2). Calculate f ^-1(x)

Solution Since f(x) = 1/2(x + 2)

... y = 1/2(x + 2)

2y = x + 2

... x = 2y - 2

Interchanging x for y, (Always remember this step; it must also be explicitly stated.)

... y = 2x - 2

... f ^-1(x) = 2x - 2

Example 2: Given f(x) = 1/2x and g(x) = x - 2

Calculate:

(i) g(-2)

(ii)fg(4)

(iii)f -1(4)

Solution (i) Since g(x) = x - 2, then g(-2) = -2 -2 = -4

N.B. x is replaced by -2 in g(x).

(ii) Given g(x) = x - 2

then fg (x) = f(x - 2)

... fg (4) = f(4 - 2) = f(2)

As f(x) = 1/2x ... f(2) = 2/2 = 1.

I am sure that you can now show that fg (x) = (x - 2)/2

(iii) As f(x) = 1/2x

then y = x/2

... x = 2y. Interchanging x for y

... y = 2x

... f ^-1(x) = 2x

... f ^-1(4) = 8

Please be sure that you are comfortable with the methods of cross multiplication and changing the subject of a formula.

Inverse of a composite function

Given the functions y = f(x) and y = g(x), then y = gf(x) is a composite function.

Since gf(x) is a function of x, the inverse is found by using the method outlined above.

Example: Given the functions f(x) = 3x and g(x) = x - 2, determine the functions:

(a) fg(x)

(b) fg ^-1(x)

Solution:

(a) As f(x) = 3x and g(x) = x - 2

... fg(x) = f(x - 2) = 3(x - 2)

... fg(x) = 3(x - 2)

(b) y = fg(x) = 3(x - 2)

... y = 3x - 6

... 3x = y + 6

... x = (y + 6)/3 Interchange x for y

... y = (x + 6)/3

... The inverse of fg(x) OR {fg}^-1 (x) is (x + 6)/3

Let us attempt another example:

Example

Given f(x) = x² and g(x) = 5x + 3, calculate

(i) f(-2)

(ii) gf(-2)

(iii) (g f) ^-1x .

Solution

(i) Since f(x) = x² ... f(-2) = (-2)2 = 4.

Answer: f(-2) = 4.

(ii) As seen from above, f(-2) = 4

And since g(x) = 5x + 3 ... gf(-2) = g(4) = (5 x 4) + 3 = 23

... gf(-2)= 23

(iii) Given that f(x) = x² and g(x) = 5x + 3

then gf(x) = g(x²)

Since g(x) = 5x + 3 ... g(x²) = 5x² + 3

... gf(x) = 5x² + 3

(NB. If gf(x) = 5x² + 3, then gf(-2) = 5 x (-2)2 + 3 = 23 as above.)

In order to find the inverse, then let y = gf(x)

... y = 5x² + 3

... 5x² = y - 3

... x² = (y - 3)/5

... x = square root of (y - 3)/5

Interchanging x for y

... y = square root of (x - 3)/5

... (g f)^-1 x = square root of (x - 3)/5

Please do the following for homework.

• f and g are functions defined as follows

f : x --> (x + 1)/2

g : x --> 2x + 7

(a) Calculate the value of f (-3)

(b) Write expressions for (i) f -1(x)

(ii) g -1 (x)

(c) Hence, or otherwise, write an expression for (gf) -1

Given that f : x --> x + 3 and g : x ¨ 2x

(a) Determine fg^-1(x) and g-1 f-1 (x)

(b) Hence evaluate fg^-1 (5 ) and g^-1 f^-1 (5)

Have a good week.

In this file photo, former prime minister, Edward Seaga (right) in disucssion with (from left) Jonoi Ramsay, Warren Gordon, Hartley Neita and R. Danny Williams, chairman, board of governors of Jamaica College on the school compound. - Rudolph Brown/Chief Photographer

Clement Radcliffe is the principal of Glenmuir High School in May Pen.