Solve One-to-one Function

Solve One-to-One Function

One-to-one function is a function, in which every element of the range of the function is corresponds to exactly one element of the domain of the function. One-to-one function is often written as 1–1 function.

Example:

Function: y = f(x) is a function, if it passes only the vertical line test.

One-to-function: y = f(x) is a one-to-one function, if it passes both the horizontal line test and the vertical line test.


Solve One-to-One Function – Example Problems


See these solved problems on one-to-one function.

Example 1: Show that the function f(x) = 4(x - 6)3 + 9 is one-to-one function.

Solution:

Let (x) = f(y)

4(x - 6)2 + 9 = 4(y - 6)3 + 9

Add -9 to both sides

4(x - 6)3 + 9 - 9 = 4(y - 6)3 + 9 - 9

4(x - 6)3 = 4(y - 6)3

Divide both sides by 4

(x - 6)3 = (y - 6)3

The above equation leads to two other equations

(x - 6) = (y - 6)

x = y

Therefore given function f(x) = 4(x - 6)2 + 9 is one-to-one function.

Example 2: Show that the rational function f(x) = `10 / (12x + 15)` is one-to-one function.

Solution:

Let f(x) = f(y)

`10 / (12x + 15)` = `10 / (12y + 15)`

Multiply both sides (12x + 15)(12y + 15) and simplify

12y + 15 = 12x + 15

Add -15 to both sides

12y = 12x

Divide both sides by 12

y = x

Therefore the given function f(x) = `10 / (12x + 15)` is one-to-one function.

Example 3: Show that the function f(x) = 15x + 18, is one-to-one function.

Solution:

Let f(x) = f(y) and show that this leads to x = y

15x + 18 = 15y + 18

Add -18 to both sides

15x = 15y

Divide both sides by 15

x = y

Therefore the given function f(x) = 15x + 18 is one-to-one.

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Solve One-to-One Function – Practice Problems


Solve these following practice problems

Problem 1: Show that the rational function f(x) = `7 / (11x + 6) ` is one-to-one function.

Problem 2: Show that the function f(x) = 5(x - 11)3 + 21 is one-to-one function.

Problem 3: Show that function f(x) = 19x + 4 is one-to-one function.