functions
for problems 1–6, determine whether each relation represents a function. explain.
1
2
3 inputs outputs
-2
1 7
14 16
4
x y
2 3
7 -8
-1 11
2 3
5 {(-2,6), (0, 0), (2, -6), (4, -12), (0, -18)}
6 {(-8, 4), (-2, 1), (3, 4), (8, 1), (9, 17)}
7 why does $y = x^2$ represent a function, but $x = y^2$ does not when $x$ is the input and $y$ is the output?

Question

Accepted Answer

### Problem 1:
# Explanation:
## Step 1: Recall the definition of a function.
A relation is a function if every input (x - value) has exactly one output (y - value). For a graph, we can use the vertical line test: if any vertical line intersects the graph at more than one point, it is not a function.
## Step 2: Apply the vertical line test to the graph.
Looking at the first graph, if we draw a vertical line at any x - value, it will intersect the graph (the line and the point) at most once. The point and the line do not violate the vertical line test because the point is above the line, and for each x, there is only one y.
# Answer:
This relation is a function. Because for every x - value, there is exactly one y - value (passes the vertical line test).

### Problem 2:
# Explanation:
## Step 1: Recall the vertical line test.
A graph represents a function if no vertical line intersects the graph more than once.
## Step 2: Apply the vertical line test to the circle graph.
For a circle centered at the origin, if we draw a vertical line (for example, $x = 0$ or $x = 1$ or $x=- 1$), the vertical line will intersect the circle at two points. So, it fails the vertical line test.
# Answer:
This relation is not a function. Because a vertical line intersects the circle (the graph) at more than one point (fails the vertical line test).

### Problem 3:
# Explanation:
## Step 1: Recall the function definition for mappings.
A mapping from inputs (domain) to outputs (range) is a function if each input is mapped to exactly one output.
## Step 2: Analyze the mapping.
The input $1$ is mapped to $- 2$, $7$, and $16$. So, the input $1$ has more than one output.
# Answer:
This relation is not a function. Because the input $1$ is mapped to multiple outputs.

### Problem 4:
# Explanation:
## Step 1: Recall the function definition for tables.
A table represents a function if each $x$ - value (input) is associated with exactly one $y$ - value (output).
## Step 2: Analyze the table.
The $x$ - value $2$ is associated with $y = 3$ in both the first and the fourth rows. But wait, actually, in a function, we allow the same output for different inputs, but we need each input to have only one output. Here, the input $2$ has only one output ($y = 3$) (even though it appears twice, it's the same output for the same input). Wait, no, let's check again. The $x$ values are $2$, $7$, $-1$, $2$. The $x$ value $2$ is paired with $y = 3$ both times. But in terms of function definition, a function can have the same output for different inputs, and for a given input, there is only one output. So, for $x = 2$, the output is $3$ (only one output), for $x=7$ output is $- 8$, for $x=-1$ output is $11$. So each $x$ has exactly one $y$.
# Answer:
This relation is a function. Because each $x$ - value is associated with exactly one $y$ - value.

### Problem 5:
# Explanation:
## Step 1: Recall the function definition for ordered pairs.
A set of ordered pairs $(x,y)$ represents a function if each $x$ - value is paired with exactly one $y$ - value.
## Step 2: Analyze the ordered pairs.
The $x$ - value $0$ is paired with $y = 0$ and $y=-18$. So, the input $0$ has more than one output.
# Answer:
This relation is not a function. Because the $x$ - value $0$ is paired with two different $y$ - values.

### Problem 6:
# Explanation:
## Step 1: Recall the function definition for ordered pairs.
A set of ordered pairs $(x,y)$ is a function if each $x$ is paired with exactly one $y$.
## Step 2: Analyze the ordered pairs.
For $x=-8$, $y = 4$; $x=-2$, $y = 1$; $x = 3$, $y = 4$; $x = 8$, $y = 1$; $x = 9$, $y = 17$. Each $x$ - value is paired with exactly one $y$ - value (note that different $x$ - values can have the same $y$ - value, which is allowed in a function).
# Answer:
This relation is a function. Because each $x$ - value is paired with exactly one $y$ - value.

### Problem 7:
# Explanation:
## Step 1: Recall the function definition (input - output).
A relation is a function when, for each input (x), there is exactly one output (y).
## Step 2: Analyze $y=x^{2}$.
For any real number $x$ (input), when we calculate $y=x^{2}$, we get exactly one real number $y$ (output). For example, if $x = 2$, $y=4$; if $x=-2$, $y = 4$ (it's okay to have the same output for different inputs, we just need one output per input).
## Step 3: Analyze $x = y^{2}$ with $x$ as input and $y$ as output.
If we take an input $x$ (for example, $x = 4$), then $y^{2}=4$ implies $y = 2$ or $y=-2$. So, for the input $x = 4$, there are two possible outputs ($y = 2$ and $y=-2$). Thus, it does not satisfy the function definition (one input has more than one output).
# Answer:
$y = x^{2}$ is a function because for every input $x$, there is exactly one output $y$ (e.g., $x = 2$ and $x=-2$ both give $y = 4$, but each $x$ has one $y$). $x=y^{2}$ (with $x$ as input, $y$ as output) is not a function because for an input $x>0$ (e.g., $x = 4$), there are two outputs ($y = 2$ and $y=-2$), so one input has multiple outputs.

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Problem 1:

Step 1: Recall the definition of a function.

Step 2: Apply the vertical line test to the graph.

Step 1: Recall the vertical line test.

Step 2: Apply the vertical line test to the circle graph.

Step 1: Recall the function definition for mappings.

Step 2: Analyze the mapping.

Answer:

Problem 2: