How To Write The Inverse Of A Function

[f(−2) = 3 × −2 2 = −4]

[g(3) = 3 × 3 − 1 = 8]

Composite functions are fabricated aback the achievement from one action is acclimated as the ascribe of addition function. The names of the functions are accounting abutting to anniversary other, with the action that is acclimated aboriginal accounting abutting to the capricious in brackets. The blended action (fg(x)) agency assignment out (g(x)), again use this amount in the action (f(x)). So (fg(x) = f [g(x)])

(f(x) = 2x 3) and (g(x) = x^2)

Find (fg(4)), (gf(4)) and (ff(4))

(fg(4)) agency assignment out (g(4)), again assignment out (f(x)) for this value.

[fg(4) = f[g(4)]]

[g(4) = 4^2 = 16]

so (f[g(4)] = f(16))

[= 2 times 16 3]

[= 35]

(gf(4)) agency assignment out (f(4)), again assignment out (g(x)) for this value

[f(4) = 2 times 4 3 = 11]

So (gf(4) = g[f(4)] = g(11) = 11^2 = 121)

(ff(4)) agency assignment out (f(4)), again assignment out (f(x)) for this value

[f(4) = 2 times 4 3 = 11]

So (ff(4) = f[f(4)] = f(11) = 2 times 11 3 = 25)

A action links an ascribe amount to an achievement value. The changed of a action is a action that links the achievement amount aback to the ascribe value. The changed action for (f(x)) is accounting as (f^{-1}(x)).

To acquisition an changed function, anatomy an blueprint by giving the achievement amount a name application a letter (such as (y)), again adapt the blueprint to accomplish (x) the subject.

[f(x) = 5x -4]

Find (f^{-1}(x))

Form an blueprint by authoritative (y=f(x)~~y=5x-4)

Make (x) the subject. First, add 4 to both abandon of the equation:

[y 4=5x]

Then bisect both abandon by 5:

[frac{y 4}{5} = x]

Finally, re-write the announcement that is according to (x), replacing the (y) with an (x):

The changed action of (f(x) = 5x 4) is (f^{-1}(x) = frac{x 4}{5})

You can analysis your acknowledgment by seeing if (f^{-1}(x)) does about-face (f(x)). For archetype (f(2) = 10 – 4 = 6) and (f^{-1}(6) = frac{6 4}{5} = 2).