If the function itself is considered a "DO" action, then the inverse is the "UNDO"

What about the domain and Range?

"Mathy" Answer

Definition: The inverse of a function is when the domain
and the range trade places. All elements of the domain become the range, and all elements of the range become the domain.

Therefore, the inverse of a function is equivalen to what kind of transformation

In this case, f(x) is a function and f^{-1}(x) is also a function.

This function passes the horizontal line test so, its inverse will also be a function.

Practice Problems I

Problem 1)
f(x) = { (1,2) , (3,4) , (5,6) }
What is the inverse of this function?

Answer

Remember: $$ f^{-1} (x) $$ means 'the inverse of f(x) '
$$ f^{-1} (x) $$ = { (2,1) , (4,3) , (6,5) }

Load More Problems Like This

Problem 2)
f(x) = { (33,14) , (23,15) , (11,12), (13, ,14 )}
What is the inverse of this function?

Answer

Remember: $$ f^{-1} (x) $$ means 'the inverse of f(x) '
$$ f^{-1} (x) $$ = { (14,33) , (15, 23) , (12,11) , (14, 13) }}

Problem 3)
f(x) = { (-1, 12) , (13, 114) , (15,61), ( 1, 12)}
What is the inverse of this function?

Answer

Remember: $$ f^{-1} (x) $$ means 'the inverse of f(x) '
$$ f^{-1} (x) $$ = { (12, -1) , (114, 13) , (61,15) , ( 12,1) }

Problem 4)
What is the inverse of y = 3

Answer

y = 3 is a horizontal line. In a horizontal line all elements of the range are 3.
So, it looks something $$ f^{-1} (x) $$ ={ ( -1,3 ) , (0,3) , (1,3)}

Therefore the inverse of this function will be whatever line has 3 for all elements in its
domain.

Therefore the inverse of y =3 is the line x = 3

Solving algebraically for the Inverse

What is the inverse of f(x) = x +1?

Just like in our prior examples, we need to switch the domain and range. In an equation , the domain is represented by the
x variable and the range by the
y variable

Show Me the Steps

To Summarize

f(x): took an element from the domain and added 1 to arrive at the corresponding element in the range.

f^{-1}(x) : took an element from the domain and subtracted 1 to arrive at the corresponding element in the range.

Practice Problems II

Problem 1)
What is the inverse of the function f(x) = x + 22?

Step 1) replace f(x) with y

y = x + 22

Step2) switch x and y

x = y + 22

Step 3) solve for new 'y'

Step 4) replace 'y' with f^{-1}(x)

f^{-1}(x) = x - 22

Problem 2)
What is the inverse of the function f(x) = 2x ?

Step 1) replace f(x) with y

y = 2x

Step2) switch x and y

x = 2y

Step 3) solve for new 'y'

$$
\frac{1}{2} x = \frac{1}{2} \cdot 2y
\\
\frac{1}{2} x = y
$$

Step 4) replace 'y' with f^{-1}(x)

$$
f^{-1 } (x) =\frac{1}{2} x
$$

Problem 3)
What is the inverse of the function $$ f(x) = \frac{1}{2}x + 3 $$ ?

Step 1) replace f(x) with y

$$ y = \frac{1}{2}x + 3 $$

Step2) switch x and y

$$ x = \frac{1}{2}y + 3 $$

Step 3) solve for new 'y'

$$
\color{Red}{2 \cdot } x = \color{Red}{2 \cdot } \big ( \frac{1}{2}y + 3 \big)
\\
2 \cdot x = \frac{2}{2}\cdot y + 2 \cdot 3
\\
2x = y + 6
\\
2x -6 = y
$$

Step 4) replace 'y' with f^{-1}(x)

$$
f^{-1 } (x) = 2x -6
$$

Problem 4)
What is the inverse of the function $$ f(x) = x^2 + 3 $$ ?

Step 1) replace f(x) with y

$$
y = x^2 + 3
$$

Step2) switch x and y

$$
x = y^2 + 3
$$

Step 3) solve for new 'y'

$$
x \color{Red}{ -3} = y^2 + 3 \color{Red}{ -3}
\\
x -3 = y^2
\\
\sqrt{x-3} = \sqrt{y^2}
\\
\sqrt{x-3} = y
$$

Step 4) replace 'y' with f^{-1}(x)

$$
f^{-1 } (x) = \sqrt{x-3}
$$

Problem 5)
What is the inverse of the function $$ f(x) = 5 \sqrt{ x } $$ ?