Question

1. How do you determine whether a function is an inverse of another function?
Add the functions.
Multiply the functions.
Find the composite of the functions.
Apply the vertical line test.

2. Which of the following is the inverse function of f(x) = 3x?

f(x) = x + 3


f(x) = x/3


f(x) = x - 3


f(x) = x3

3. Which of the following statements is true?

A function will always pass the vertical line test.


All the answers are correct.


If the function has an inverse function, then the inverse function will pass the vertical line test.


If a function has an inverse function, then the original function will pass the horizontal line test.

4. What is the inverse function of f(x) = 3(x - 2)3?




5. Which of the following is the inverse function of f(x) = 2x - 3?

f(x) = (2x - 3 + 3)/2


f(x) = x+ 3/2


f(x) = x/2 + 3


f(x) = (x+3)/2

Answer 1

Answer:

1. Find the composite of the functions
2. x/3
3. All the answers are correct
4. f(x) = 2+∛(x/3)
5. f(x) = (x+3)/2

Step-by-step explanation:

1. If f(x) and g(x) are inverse functions, then f(g(x)) = g(f(x)) = x. Finding the composite of the two functions will tell you if they are inverses.

2. To find the inverse of a function, swap x and y, then solve for y.

... x = 3y

... x/3 = y . . . . . matches f(x) = x/3

3. A function will pass the vertical line test. If its inverse is also a function, that, too, will pass the vertical line test. Since the inverse of a function is that function reflected across y=x, any inverse function that passes the vertical line test corresponds to an original function that passes the horizontal line test. (A vertical line reflected across y=x is a horizontal line.)

4. See 2.

... x = 3(y -2)³

... (x/3) = (y -2)³ . . . . divide by 3

... ∛(x/3) = y -2 . . . . .take the cube root

... 2+∛(x/3) = y . . . . .add 2

... f(x) = 2+∛(x/3) . . . . is the inverse

5. See 2.

... x = 2y -3

... x+3 = 2y . . . . . add 3

... (x+3)/2 = y . . . .divide by 2

... f(x) = (x+3)/2 . . . .  is the inverse

Answer 2

A composite function is the combination of multiple functions.

The correct answers are:

1. Find the composite of the functions.
2. The inverse of f(x) = 3x is $f'(x) = \frac x3$.
3. All answers are true
4. The inverse of $f(x) = 3(x - 2)^3$ is: $f^{-1}(x) =2 + \sqrt[3]{\frac x3}$
5. The inverse of $f(x) =2x - 3$ is $f^{-1}(x) = \frac{x + 3}{2}$

1. Test for inverse function

To test if two functions are inverse of one another, we simply find their composites.

Assume the functions are g(x) and h(x).

We simply test for $g(h^{-1}(x))$ and $h(g^{-1}(x))$

If they are equal, then both functions are inverse functions

2. Inverse of f(x) = 3x

Rewrite as:

$y = 3x$

Swap y and x

$x = 3y$

Make y the subject

$y = \frac x3$

Hence, the inverse function is: $f'(x) = \frac x3$

3. True statements

A function has unique ordered pairs; so, it will pass the vertical line test.

Because it has unique ordered pairs, the inverse function will pass the vertical line tests, and the horizontal line tests.

Hence;

(b) All answers are correct

4. Inverse of $f(x) = 3(x - 2)^3$

Rewrite as:

$y = 3(x - 2)^3$

Swap x and y

$x = 3(y - 2)^3$

Solve for y: Divide both sides by 3

$(y -2)^3 = \frac x3$

Take cube roots of both sides

$y -2 = \sqrt[3]{\frac x3}$

Add 2 to both sides

$y =2 + \sqrt[3]{\frac x3}$

Hence, the inverse function is: $f^{-1}(x) =2 + \sqrt[3]{\frac x3}$

5. The inverse of $f(x) =2x - 3$

Rewrite as:

$y =2x - 3$

Swap x and y

$x =2y - 3$

Solve for y: Add 3 to both sides

$2y = x + 3$

Divide both sides by 2

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

Hence, the inverse function is: $f^{-1}(x) = \frac{x + 3}{2}$

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