Digital SAT Maths -Unit 2 - 2.9 Inverse Functions- Study Notes- New Syllabus
Digital SAT Maths -Unit 2 – 2.9 Inverse Functions- Study Notes- New syllabus
Digital SAT Maths -Unit 2 – 2.9 Inverse Functions- Study Notes – per latest Syllabus.
Key Concepts:
Finding inverse
Meaning of inverse
Finding the Inverse
An inverse function reverses what the original function does.
If a function takes input \( x \) and gives output \( y \), the inverse takes \( y \) back to \( x \).
\( f(x)=y \quad \Longrightarrow \quad f^{-1}(y)=x \)
Key Property
\( f(f^{-1}(x))=x \)
Steps to Find an Inverse
- Write \( y=f(x) \)
- Swap \( x \) and \( y \)
- Solve for \( y \)
- Rename \( y \) as \( f^{-1}(x) \)
DIGITAL SAT Tip
Most SAT inverse questions use linear or simple quadratic expressions.
Example 1:
Find the inverse of \( f(x)=3x+5 \).
▶️ Answer/Explanation
\( y=3x+5 \)
Swap → \( x=3y+5 \)
\( x-5=3y \)
\( y=\dfrac{x-5}{3} \)
Answer: \( f^{-1}(x)=\dfrac{x-5}{3} \)
Example 2:
Find the inverse of \( f(x)=\dfrac{x-2}{4} \).
▶️ Answer/Explanation
\( y=\dfrac{x-2}{4} \)
Swap → \( x=\dfrac{y-2}{4} \)
\( 4x=y-2 \)
\( y=4x+2 \)
Answer: \( f^{-1}(x)=4x+2 \)
Example 3:
Given \( f(x)=2x-7 \), verify the inverse.
▶️ Answer/Explanation
Find inverse:
\( y=2x-7 \Rightarrow x=2y-7 \)
\( y=\dfrac{x+7}{2} \)
Check:
\( f(f^{-1}(x))=2\left(\dfrac{x+7}{2}\right)-7=x \)
Verified
Meaning of an Inverse
The inverse function reverses the action of the original function.
If a function converts an input into an output, the inverse converts that output back into the original input.
Original: input \( \rightarrow \) output
Inverse: output \( \rightarrow \) input
Important Property
\( f(f^{-1}(x))=x \) and \( f^{-1}(f(x))=x \)
They undo each other.
Graph Meaning
The graph of a function and its inverse are mirror images across the line:
\( y=x \)
Domain & Range Relationship
- Domain of \( f \) becomes range of \( f^{-1} \)
- Range of \( f \) becomes domain of \( f^{-1} \)
Real-Life Interpretation (Very Important for SAT)
Many SAT questions describe a process and ask you to interpret the inverse.
Example idea:
function: money spent → reward points
inverse: reward points → money spent
DIGITAL SAT Tip
If a question asks “what input produced this output?”, it is asking for the inverse.
Example 1:
A function converts temperature from Celsius to Fahrenheit. What does the inverse represent?
▶️ Answer/Explanation
Original: Celsius → Fahrenheit
Inverse: Fahrenheit → Celsius
Answer: It converts Fahrenheit back to Celsius.
Example 2:
If \( f(x) \) gives the total cost after buying \( x \) tickets, what does \( f^{-1}(x) \) represent?
▶️ Answer/Explanation
The inverse asks: how many tickets produced that total cost?
Answer: the number of tickets that correspond to a given cost.
Example 3:
A function maps study hours to test score. What does the inverse tell you?
▶️ Answer/Explanation
Inverse finds how many study hours produced a particular score.
Answer: the study time required to achieve a certain score.