Solved Worksheet- 5.2-One-to-One and Inverse Functions (1) | Chegg.com - Free Printable

Let's go through each problem step by step.

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(1) If a function is one-to-one, what does that tell you about the inverse of the function?

Answer:
If a function is one-to-one, then its inverse is also a function.

Explanation:
A function is one-to-one if every output (y-value) corresponds to exactly one input (x-value). This means no two different inputs produce the same output. Because of this, when we reverse the inputs and outputs to form the inverse, each input of the inverse will have only one output — satisfying the definition of a function.

So, a one-to-one function has an inverse that is also a function.

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(2) Determine whether the function is one-to-one. Then tell whether its inverse is a function.

Given:
- Domain: {-5, -2, 0, 3, 4}
- Range: {-1, 5, 7, 12, 21}
- Mappings:
- -5 → 2
- -2 → 5
- 0 → 7
- 3 → 12
- 4 → 21

Wait — there's a typo in the diagram. The arrow from -5 goes to 2, but 2 is not in the range list. The listed range is {-1, 5, 7, 12, 21}. So this seems inconsistent.

But looking closely at the arrows:
- -5 → 2 ✘ (2 not in range)
- -2 → 5 ✔
- 0 → 7 ✔
- 3 → 12 ✔
- 4 → 21 ✔

But 2 is not in the range set — so likely a mistake.

Alternatively, perhaps it's meant to be:
- -5 → -1
- -2 → 5
- 0 → 7
- 3 → 12
- 4 → 21

But the arrow from -5 points to 2, which is not in the range.

Wait — maybe the range values are mislabeled?

Looking again:
The right oval says "Range" with values: -1, 5, 7, 12, 21
But the arrow from -5 goes to 2, which is not listed.

This is confusing. But let's assume it's a typo and the mapping is:

- -5 → -1
- -2 → 5
- 0 → 7
- 3 → 12
- 4 → 21

Then all outputs are in the range, and each x maps to a unique y, and no two x-values map to the same y.

So check for one-to-one:
- All y-values are distinct.
- No repeated outputs.

✔ So it is one-to-one.

Therefore, the inverse is a function.

Now, find domain and range of the inverse:

- Inverse swaps domain and range.
- Original domain: {-5, -2, 0, 3, 4} → becomes range of inverse
- Original range: {-1, 5, 7, 12, 21} → becomes domain of inverse

So:
- Domain of inverse: {-1, 5, 7, 12, 21}
- Range of inverse: {-5, -2, 0, 3, 4}

> Answer (2):
Yes, the function is one-to-one.
Its inverse is a function.
Domain of inverse: {-1, 5, 7, 12, 21}
Range of inverse: {-5, -2, 0, 3, 4}

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(3) Table:

| x | y |
|---|---|
| -5 | -1 |
| -2 | 0 |
| 5 | 4 |
| 8 | -1 |

Check if one-to-one:

Look at y-values: -1, 0, 4, -1

→ -1 appears twice (for x = -5 and x = 8)

So two different inputs give same output → not one-to-one

Thus, the inverse would have two x-values mapping to the same y, so inverse is not a function

> Answer (3):
Not one-to-one.
Inverse is not a function.

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(4) Set of ordered pairs: {(-7,6), (-4,-2), (-3,6), (-4,-2)}

Wait — duplicate points? Let's list them:

- (-7, 6)
- (-4, -2)
- (-3, 6)
- (-4, -2) → already listed

So actually: {(-7,6), (-4,-2), (-3,6)}

Wait — but (-4,-2) appears twice — irrelevant.

Now check if one-to-one:

- x = -7 → y = 6
- x = -4 → y = -2
- x = -3 → y = 6

So both -7 and -3 map to 6 → same output for different inputs

→ Not one-to-one

So inverse is not a function (since y=6 would map to both x=-7 and x=-3)

> Answer (4):
Not one-to-one.
Inverse is not a function.

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(5) f(x) = x² - 5

This is a parabola opening upward.

Is it one-to-one?

No. For example:
- f(2) = 4 - 5 = -1
- f(-2) = 4 - 5 = -1

Same output for different inputs → not one-to-one

So inverse is not a function

> Answer (5):
Not one-to-one.
Inverse is not a function.

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(6) Graph: increasing curve from (-3, -1) to (3, 1)

Looks like part of a square root or logarithmic function, but it's strictly increasing.

It passes the horizontal line test: any horizontal line crosses the graph at most once.

So it is one-to-one

Therefore, inverse is a function

Now, find domain and range of inverse:

Original function:
- Domain: [-3, 3] (from graph)
- Range: [-1, 1]

So inverse:
- Domain: [-1, 1]
- Range: [-3, 3]

> Answer (6):
One-to-one.
Inverse is a function.
Domain of inverse: [-1, 1]
Range of inverse: [-3, 3]

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(7) Graph: V-shaped, vertex at (0,0), arms going up to left and right

This is f(x) = |x|

Check if one-to-one:

- f(2) = 2
- f(-2) = 2 → same output

So not one-to-one

→ Inverse is not a function

> Answer (7):
Not one-to-one.
Inverse is not a function

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(8) Graph of f(x) given; sketch f⁻¹(x) on same axes

We need to reflect the graph of f(x) over the line y = x to get f⁻¹(x).

Steps:
1. Identify key points on f(x)
2. Swap x and y coordinates
3. Plot those points
4. Connect smoothly

From the graph:

Let’s estimate key points:

- From (-4, -3) → point on f(x)
- (-2, -2)
- (0, -1)
- (1, 0)
- (2, 1)
- (3, 2)
- (4, 4)

Wait — the graph looks like:
- A straight line from (-4, -3) to (0, -1) → slope = (-1 + 3)/(0 + 4) = 2/4 = 0.5
- Then from (0, -1) to (1, 0) → slope = 1
- Then from (1, 0) to (4, 4) → slope = 4/3 ≈ 1.33

But more carefully:

Let’s pick exact points:

- (-4, -3)
- (-2, -2)
- (0, -1)
- (1, 0)
- (2, 1)
- (3, 2)
- (4, 4)

Now, to get f⁻¹(x), swap x and y:

- (-3, -4)
- (-2, -2)
- (-1, 0)
- (0, 1)
- (1, 2)
- (2, 3)
- (4, 4)

Now plot these points and connect them in order.

Note: The original graph has three segments:
1. From (-4, -3) to (0, -1): linear
2. From (0, -1) to (1, 0): linear
3. From (1, 0) to (4, 4): linear

So inverse will have:
1. From (-3, -4) to (-1, 0): this is reflection of first segment
2. From (-1, 0) to (0, 1): reflection of second
3. From (0, 1) to (4, 4): reflection of third

Plot these reflected points and draw lines between them.

Also, note that since f(x) is strictly increasing (as seen from graph), it is one-to-one, so inverse exists and is a function.

To graph f⁻¹(x), draw the reflection over y = x.

You can do this by:
- Drawing the line y = x
- Reflecting each point across it

So final answer for (8):

> Answer (8):
Since f(x) is one-to-one, f⁻¹(x) exists. To graph it, reflect the graph of f(x) across the line y = x. Use the swapped points as above and connect them with appropriate line segments.

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✔ Final Answers Summary:

(1)
If a function is one-to-one, then its inverse is a function.

(2)
One-to-one: Yes
Inverse is a function: Yes
Domain of inverse: {-1, 5, 7, 12, 21}
Range of inverse: {-5, -2, 0, 3, 4}

(3)
One-to-one: No
Inverse is a function: No

(4)
One-to-one: No (since (-7,6) and (-3,6))
Inverse is a function: No

(5)
One-to-one: No
Inverse is a function: No

(6)
One-to-one: Yes
Inverse is a function: Yes
Domain of inverse: [-1, 1]
Range of inverse: [-3, 3]

(7)
One-to-one: No
Inverse is a function: No

(8)
Graph f⁻¹(x) by reflecting f(x) over the line y = x.
Use key points:
Original: (-4,-3), (-2,-2), (0,-1), (1,0), (2,1), (3,2), (4,4)
Inverse: (-3,-4), (-2,-2), (-1,0), (0,1), (1,2), (2,3), (4,4)
Connect with line segments.

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Let me know if you'd like a visual sketch description or help plotting (8).