unit review a: foundations for funct problems 6 - 9 find the inverse of each relation. determine if the relation and its inverse are functions or not. 6. find the inverse of the relation.
| x | y |
| -4 | 4 |
| 3 | 5 |
| 4 | 6 |
| 5 | 7 |
| 12 | 8 |
is the relation a function? is its inverse a function? 7. find the inverse of the relation. (diagram of two ovals with arrows: -5→7, -5→8, -4→7, -4→8, -4→9, -1→10) is the relation a function? is its inverse a function? 8. find the inverse of the relation. {(5, -1), (5, -3), (5, -6), (4, 10)} is the relation a function? is its inverse a function? find the domain and range of the relation. find the domain and range of its inverse. 9. graph the inverse of the relation on the same grid. (graph of a curve) is the relation a function? is its inverse a function?

Question

Accepted Answer

### Problem 6
# Explanation:
## Step1: Find the inverse relation
To find the inverse of a relation given by a table, we swap the $x$ and $y$ values. The original relation has pairs $(-4, 4)$, $(3, 5)$, $(4, 6)$, $(5, 7)$, $(12, 8)$. Swapping $x$ and $y$, the inverse relation is $(4, -4)$, $(5, 3)$, $(6, 4)$, $(7, 5)$, $(8, 12)$.

## Step2: Check if the original relation is a function
A relation is a function if each $x$-value has exactly one $y$-value. In the original table, each $x$ ( -4, 3, 4, 5, 12) has a unique $y$, so it is a function.

## Step3: Check if the inverse is a function
For the inverse relation, each $x$-value (4, 5, 6, 7, 8) has exactly one $y$-value, so the inverse is also a function.

# Answer:
Inverse relation: $\{(4, -4), (5, 3), (6, 4), (7, 5), (8, 12)\}$
Original relation: Yes (it is a function)
Inverse relation: Yes (it is a function)

### Problem 7
# Explanation:
## Step1: Find the inverse relation
The original relation is a mapping from the left oval (domain) to the right oval (range). To find the inverse, we reverse the arrows. So, 7 maps to -5, 8 maps to -5 and -4, 9 maps to -4, 10 maps to -1. So the inverse relation has pairs $(7, -5)$, $(8, -5)$, $(8, -4)$, $(9, -4)$, $(10, -1)$.

## Step2: Check if the original relation is a function
A function has each domain element mapping to exactly one range element. Here, -5 maps to 7, -4 maps to 8 and 9, -1 maps to 10. Wait, no—wait, the original relation: left oval is domain, right is range. Each element in the domain (-5, -4, -1) maps to: -5 → 7; -4 → 8, 9; -1 → 10. Wait, no, a function requires each domain element to have exactly one range element. But -4 maps to two elements (8 and 9), so the original relation is **not** a function? Wait, no, wait the diagram: left oval has -5, -4, -1. -5 points to 7; -4 points to 8 and 9; -1 points to 10. So domain elements: -5 (1 mapping), -4 (2 mappings), -1 (1 mapping). So original relation is not a function.

Wait, I made a mistake earlier. Let's correct:

## Step1 (corrected): Original relation as a function
A function must have each domain (left) element map to exactly one range (right) element. Here, -4 maps to 8 and 9 (two elements), so the original relation is **not** a function.

## Step2: Inverse relation as a function
The inverse relation: domain is {7, 8, 9, 10}, range is {-5, -4, -1}. 7 maps to -5; 8 maps to -5 and -4; 9 maps to -4; 10 maps to -1. So 8 maps to two elements (-5 and -4), so the inverse relation is also not a function.

# Answer:
Inverse relation: $\{(7, -5), (8, -5), (8, -4), (9, -4), (10, -1)\}$
Original relation: No (not a function, since -4 maps to two values)
Inverse relation: No (not a function, since 8 maps to two values)

### Problem 8
# Explanation:
## Step1: Find the inverse relation
The original relation is $\{(5, -1), (5, -3), (5, -6), (4, 10)\}$. To find the inverse, swap $x$ and $y$: $\{(-1, 5), (-3, 5), (-6, 5), (10, 4)\}$.

## Step2: Check if original relation is a function
A function has each $x$-value with exactly one $y$-value. Here, $x = 5$ maps to -1, -3, -6 (three values), so the original relation is **not** a function.

## Step3: Check if inverse is a function
The inverse relation: $x = -1$ maps to 5, $x = -3$ maps to 5, $x = -6$ maps to 5, $x = 10$ maps to 4. Each $x$-value in the inverse has exactly one $y$-value (even though multiple $x$s map to 5, that's allowed in a function—functions can have multiple inputs to the same output, just not multiple outputs from the same input). So the inverse relation **is** a function.

## Step4: Domain and range of original relation
Domain (set of $x$-values): $\{5, 4\}$ (since 5 repeats, we list it once? Wait, no—domain is all unique $x$-values. So $x = 5$ (appears three times) and $x = 4$, so domain is $\{4, 5\}$. Range (set of $y$-values): $\{-1, -3, -6, 10\}$ (all unique $y$s).

## Step5: Domain and range of inverse relation
Domain of inverse is the range of original: $\{-1, -3, -6, 10\}$. Range of inverse is the domain of original: $\{4, 5\}$.

# Answer:
Inverse relation: $\{(-1, 5), (-3, 5), (-6, 5), (10, 4)\}$
Original relation: No (not a function, $x = 5$ has multiple $y$s)
Inverse relation: Yes (it is a function, each $x$ has one $y$)
Domain of original: $\{4, 5\}$
Range of original: $\{-1, -3, -6, 10\}$
Domain of inverse: $\{-1, -3, -6, 10\}$
Range of inverse: $\{4, 5\}$

### Problem 9
# Explanation:
To graph the inverse of a relation, we reflect each point $(x, y)$ over the line $y = x$, which means swapping the $x$ and $y$ coordinates of each point on the original graph.

## Step1: Identify points on the original graph
The original graph appears to have a curve starting at some point (let's assume the visible part: for example, if the original graph has a point $(a, b)$, the inverse will have $(b, a)$.

## Step2: Check if original relation is a function
Using the vertical line test: if any vertical line intersects the graph more than once, it's not a function. The original graph (as shown) looks like a curve that passes the vertical line test (since it's a horizontal-like curve but wait, the graph in the image: let's see, the original graph—if it's a curve that, for example, has a horizontal tangent or is increasing with each $x$ having one $y$, then it's a function. Wait, the graph shown: it's a curve that starts at a point on the $x$-axis (maybe $(-2, 0)$ or similar) and increases to the right. So vertical line test: any vertical line will intersect it once, so original relation is a function.

## Step3: Check if inverse is a function
Using the horizontal line test on the original graph: if any horizontal line intersects the graph more than once, the inverse is not a function. The original graph (as shown) looks like it might fail the horizontal line test (if it's a curve that, for example, is concave or has a horizontal segment, but from the image, it's a curve that, when reflected, would we have multiple $x$s for a $y$? Wait, the original graph: let's assume it's a square root-like graph (starting at $(h, 0)$ and increasing). For a square root function $y = \sqrt{x - h}$, the inverse is $x = \sqrt{y - h}$ → $y = (x)^2 + h$ (for $x \geq 0$), which is a parabola opening upwards, which fails the horizontal line test (so inverse is not a function). Wait, but let's check:

Original graph: if it's a function (passes vertical line test), then inverse is a function only if original passes horizontal line test (one-to-one). The original graph shown: does it pass horizontal line test? If the original graph is, say, $y = \sqrt{x + 2}$ (starting at $(-2, 0)$ and increasing), then a horizontal line $y = k$ (for $k \geq 0$) intersects it once, so it is one-to-one, so inverse is a function. Wait, maybe the graph is a function that is one-to-one.

Wait, the problem: "Is the relation a function? Is its inverse a function?"

Original graph: vertical line test—if the graph is as shown (a curve that doesn't have two points with the same $x$), so it's a function.

Inverse graph: after reflecting over $y = x$, we check vertical line test on the inverse (which is horizontal line test on original). If original is one-to-one (passes horizontal line test), then inverse is a function.

But from the image, the original graph: let's see, the curve is increasing, so each $y$ has one $x$, so original is one-to-one, so inverse is a function. Wait, maybe I'm overcomplicating.

# Answer:
To graph the inverse, swap $x$ and $y$ for each point on the original graph.
Original relation: Yes (it is a function, passes vertical line test)
Inverse relation: No (if original graph fails horizontal line test, but from the image, if it's a square root-like graph, maybe yes? Wait, no—wait the original graph: if it's a curve that, when reflected, the inverse would be a parabola, which fails horizontal line test. Wait, maybe the original graph is a function but not one-to-one, so inverse is not a function.

But based on the image:

Original relation: Yes (function, vertical line test passes)
Inverse relation: No (since original graph likely fails horizontal line test, so inverse is not a function)

(Note: The exact graph's points are not fully clear, but the process is: reflect over $y=x$, check vertical/horizontal line tests.)

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Problem 6

Step1: Find the inverse relation

Step2: Check if the original relation is a function

Step3: Check if the inverse is a function

Step1: Find the inverse relation

Step2: Check if the original relation is a function

Step1 (corrected): Original relation as a function

Step2: Inverse relation as a function

Step1: Find the inverse relation

Step2: Check if original relation is a function

Step3: Check if inverse is a function

Step4: Domain and range of original relation

Step5: Domain and range of inverse relation

Answer:

Problem 7