Inverse functions
INVERSE FUNCTIONS Instructor: Mohammad Mashayekhi
Outline: ■ One-to-one functions ■ Inverse functions ■ How to find the inverse of a function ■ How to plot an inverse function ■ Example
One-to-one functions: A function 𝑓 𝑥 is said to be one-to-one (injective) if there is only one 𝑥 value for each 𝑦 in the range of the function If 𝑥$ ≠ 𝑥& then 𝑓(𝑥$ ) ≠ 𝑓(𝑥& ) § Horizontal line test: A function 𝑓 𝑥 is one-to-one if and only if there is at most one point of intersection between the function and any horizontal line.
Inverse functions: 𝑓
§ 𝑓 B$ 𝑦 = 𝑥 is said to be the inverse of 𝑓 𝑥 = 𝑦 if: 𝑓 B$ 𝑓 𝑥
= 𝑥 ,
𝑓 𝑓 B$ 𝑦
=𝑦
A § A function has inverse if it is a one-to-one function. § Example: 𝑓 𝑥 = 𝑥 D ,
𝑓 B$ 𝑥 =
E
. x
. y
𝑥
𝑓 B$
B
Outline: ■ One-to-one functions ■ Inverse functions ■ How to find the inverse of a function ■ How to plot an inverse function ■ Example
How to find the inverse of a function: § Steps to find the inverse of a function: 1)
Put 𝑓 𝑥 equal to 𝑦
2)
Swap 𝑥 and 𝑦
3)
Solve for 𝑦
4)
Whatever you find for 𝑦 would be 𝑓 B$ (𝑥)
§ Example: Find the inverse of 𝑓 𝑥 = 2𝑥 + 1
Quiz! ■ What is 𝑓 B$ 𝑥 if 𝑓 𝑥 =
a) b) c)
&JB$ DJK$
DJK$ &JB$ &JB$ BJB$ BJB$ DJB&
d) This function is not invertible
Quiz solution: § What is 𝑓 B$ 𝑥 if 𝑓 𝑥 = Correct answer: c)
BJB$ DJB&
&JB$ DJK$
How to plot an inverse function: § Sicne we find the expression for the inverse function by swapping 𝑥 and 𝑦, we can find the plot for 𝑓 B$ 𝑥 by reflecting the plot for 𝑓(𝑥) respect to the line 𝑥 = 𝑦. § Example: 𝑓 𝑥 = 𝑥 D ,
𝑓 B$ 𝑥 =
E
𝑥
𝑥=𝑦
A non-trivial example of inverse functions: § Exponential functions and logarithmic functions are inverse of each other! 𝑙𝑜𝑔Q 𝑥 = 𝑦
■ Very useful identities:
𝑙𝑜𝑔Q 𝑎 J = 𝑥 𝑎
WXYZ J
=𝑥
↔
𝑥 = 𝑎 T 𝑓𝑜𝑟 𝑎 > 1
𝑥=𝑦
Example: § Find the inverse of the following function: 𝑓 𝑥 = −2 Ø Solution:
] J 10
𝑓𝑜𝑟 𝑥 > 0
THANKS FOR YOUR TIME!
Outline: ■ One-to-one functions ■ Inverse functions ■ How to find the inverse of a function ■ How to plot an inverse function ■ Example
One-to-one functions: A function 𝑓 𝑥 is said to be one-to-one (injective) if there is only one 𝑥 value for each 𝑦 in the range of the function If 𝑥$ ≠ 𝑥& then 𝑓(𝑥$ ) ≠ 𝑓(𝑥& ) § Horizontal line test: A function 𝑓 𝑥 is one-to-one if and only if there is at most one point of intersection between the function and any horizontal line.
Inverse functions: 𝑓
§ 𝑓 B$ 𝑦 = 𝑥 is said to be the inverse of 𝑓 𝑥 = 𝑦 if: 𝑓 B$ 𝑓 𝑥
= 𝑥 ,
𝑓 𝑓 B$ 𝑦
=𝑦
A § A function has inverse if it is a one-to-one function. § Example: 𝑓 𝑥 = 𝑥 D ,
𝑓 B$ 𝑥 =
E
. x
. y
𝑥
𝑓 B$
B
Outline: ■ One-to-one functions ■ Inverse functions ■ How to find the inverse of a function ■ How to plot an inverse function ■ Example
How to find the inverse of a function: § Steps to find the inverse of a function: 1)
Put 𝑓 𝑥 equal to 𝑦
2)
Swap 𝑥 and 𝑦
3)
Solve for 𝑦
4)
Whatever you find for 𝑦 would be 𝑓 B$ (𝑥)
§ Example: Find the inverse of 𝑓 𝑥 = 2𝑥 + 1
Quiz! ■ What is 𝑓 B$ 𝑥 if 𝑓 𝑥 =
a) b) c)
&JB$ DJK$
DJK$ &JB$ &JB$ BJB$ BJB$ DJB&
d) This function is not invertible
Quiz solution: § What is 𝑓 B$ 𝑥 if 𝑓 𝑥 = Correct answer: c)
BJB$ DJB&
&JB$ DJK$
How to plot an inverse function: § Sicne we find the expression for the inverse function by swapping 𝑥 and 𝑦, we can find the plot for 𝑓 B$ 𝑥 by reflecting the plot for 𝑓(𝑥) respect to the line 𝑥 = 𝑦. § Example: 𝑓 𝑥 = 𝑥 D ,
𝑓 B$ 𝑥 =
E
𝑥
𝑥=𝑦
A non-trivial example of inverse functions: § Exponential functions and logarithmic functions are inverse of each other! 𝑙𝑜𝑔Q 𝑥 = 𝑦
■ Very useful identities:
𝑙𝑜𝑔Q 𝑎 J = 𝑥 𝑎
WXYZ J
=𝑥
↔
𝑥 = 𝑎 T 𝑓𝑜𝑟 𝑎 > 1
𝑥=𝑦
Example: § Find the inverse of the following function: 𝑓 𝑥 = −2 Ø Solution:
] J 10
𝑓𝑜𝑟 𝑥 > 0
THANKS FOR YOUR TIME!
