Algebraic Representation of a Function - Examples, Exercises and Solutions

To determine if the graph is a function, we will use the Vertical Line Test.

The Vertical Line Test states that a graph represents a function if and only if no vertical line intersects the graph at more than one point.

Let's apply this test to the given graph, where a horizontal line is drawn. This line represents the function the graph should be verified against.

• Step 1: Conceptualize vertical lines passing through different x-values across the domain of the graph.
• Step 2: Observe if any of these vertical lines intersect the graph at more than one point.

Upon inspection of the graph, we see that every vertical line intersects the graph at exactly one point.

This indicates that for every input (x-value), there is a unique output (y-value), fulfilling the criteria for the definition of a function.

Therefore, according to the Vertical Line Test, the given graph is indeed a function.

The correct choice is: Yes

To determine whether the graph represents a function, we apply the Vertical Line Test. Here are the steps we follow:

• Step 1: Visualize placing a vertical line across various parts of the graph.
• Step 2: Check if the vertical line intersects the graph at more than one point at any given position.

Step 1: On evaluating the given graph carefully, there is a notable presence of a vertical line passing through multiple y-values. Specifically, the vertical line goes from $y = -3$ to $y = 3$ at $x = 3$.

Step 2: Since this vertical line at $x = 3$ intersects the graph at an infinite number of points, it fails the Vertical Line Test.

Therefore, the graph does not represent a function. According to our analysis and the Vertical Line Test, the correct answer is No.

To determine if the graph in question represents a function, we'll employ the Vertical Line Test. This test helps to ascertain whether each input value from the domain (x-values) is connected to a unique output value (y-values).

• According to the Vertical Line Test, a graph represents a function if no vertical line can intersect the graph at more than one point.
• In the provided diagram, the graph is a straight line.
• Visual inspection shows that any vertical line drawn at any point along the x-axis intersects the line exactly once.
• This indicates that for each x-value, there is a unique corresponding y-value. Therefore, the relationship depicted by the graph meets the criteria for a function.

Thus, the given graph correctly characterizes a function.
Therefore, the solution to the problem is Yes.

To determine if the given graph represents a function, we use the vertical line test: if any vertical line intersects the graph at more than one point, the graph is not a function.

Let's apply this test to the graph:

• Examine different sections of the graph by drawing imaginary vertical lines.
• Look for intersections where more than one point exists on the vertical line.

Upon examining the graph, we observe that there are several vertical lines that intersect the graph at multiple points, particularly in areas with loops or overlapping curves. This indicates that at those $x$-values, there are multiple $y$-values corresponding to them.

Since there exist such vertical lines, according to the vertical line test, the graph does not represent a function.

Thus, the solution to this problem is that the given graph is not a function.

It is important to remember that a function is an equation that assigns to each element in domain X one and only one element in range Y

Let's note that in the graph:

$f(0)=2,f(0)=-2$

In other words, there are two values for the same number.

Therefore, the graph is not a function.

It is important to remember that a function is an equation that assigns to each element in domain X one and only one element in range Y

We should note that for every X value found on the graph, there is one and only one corresponding Y value.

Therefore, the graph is indeed a function.

It is important to remember that a function is an equation that assigns to each value in domain $x$ only one value in range $y$.

Since we can see that for every $x$ value found on the graph there is only one corresponding$y$ value, the graph is indeed a function.

It is important to remember that a constant function describes a situation where, as the X value increases, the Y value remains constant.

In the table, we can see that there is a constant change in the X values, specifically an increase of 2, while the Y value remains constant.

Therefore, the table does indeed describe a constant function.

To solve this problem, we'll follow these steps:

• Step 1: Identify the given pairs of $(X, Y)$.
• Step 2: Verify that each $X$ maps to exactly one $Y$.
• Step 3: Conclude whether the table represents a function.

Now, let's work through each step:
Step 1: The pairs given are: $(-2, 1)$, $(2, 6)$, $(6, 11)$, $(10, 16)$, $(14, 21)$.

Step 2: For each input value $X$, we check its corresponding output $Y$:

• $X = -2$ maps to $Y = 1$
• $X = 2$ maps to $Y = 6$
• $X = 6$ maps to $Y = 11$
• $X = 10$ maps to $Y = 16$
• $X = 14$ maps to $Y = 21$

Step 3: Since each $X$ value has exactly one corresponding $Y$ value, the table represents a function.

Yes

To determine if the table represents a constant function, we need to examine the Y-values corresponding to the X-values given in the table.

• Step 1: Identify the given values from the table. The pairs are as follows: - For $X = -1$, $Y = 2$ - For $X = 0$, $Y = 4$ - For $X = 1$, $Y = 7$
• Step 2: Check if all Y-values are the same. Compare Y-values for each X-value:
- $Y = 2$ when $X = -1$, - $Y = 4$ when $X = 0$, - $Y = 7$ when $X = 1$.

Since the Y-values (2, 4, and 7) are not the same, the function is not constant.

Thus, the table does not represent a constant function. The correct choice is: No.

It is important to remember that a constant function describes a situation where as the X value increases, the function value (Y) remains constant.

In the table, we can observe that there is a constant change in X values, meaning an increase of 1, and a constant change in Y values, meaning an increase of 3

Therefore, according to the rule, the table describes a function.

It is important to remember that a constant function describes a situation where as the X value increases, the function value (Y) remains constant.

In the table, we can observe that there is a constant change in X values, meaning an increase of 1, and a non-constant change in Y values - sometimes increasing by 1 and sometimes by 4

Therefore, according to the rule, the table does not describe a function

To determine if the table represents a linear function, we need to check if the slope between each consecutive pair of points is constant.
Using the slope formula $m = \frac{y_2 - y_1}{x_2 - x_1}$, we calculate:

• Between points $(-1, 1)$ and $(2, 2)$:
  $m = \frac{2 - 1}{2 - (-1)} = \frac{1}{3}$
• Between points $(2, 2)$ and $(6, 3)$:
  $m = \frac{3 - 2}{6 - 2} = \frac{1}{4}$

Since the slopes are not equal ($\frac{1}{3} \neq \frac{1}{4}$), the function is not linear.

Thus, the table does not represent a linear function.

To determine the corresponding equation for the given table, follow these steps:

• Step 1: Confirm the linearity of the data by calculating the slope, $m$, using consecutive data points:
  - Between points $(-2, 0)$ and $(0, 1)$, the slope is $m = \frac{1 - 0}{0 - (-2)} = \frac{1}{2}$.
  - Between points $(0, 1)$ and $(2, 2)$, the slope is $m = \frac{2 - 1}{2 - 0} = \frac{1}{2}$.
  - Confirm the same slope $m = \frac{1}{2}$ for remaining pairs $(2, 2)$ and $(4, 3)$, $(4, 3)$ and $(6, 4)$.
• Step 2: Use one point, such as $(0, 1)$, to find the y-intercept, $b$, knowing the slope $m = \frac{1}{2}$:
  - Use the slope-intercept form: $y = mx + b$. Substituting $(0, 1)$, gives $1 = \frac{1}{2}(0) + b$, implying $b = 1$.
• Step 3: Formulate the equation: Given $m = \frac{1}{2}$ and $b = 1$, the linear function is:
  - $y = \frac{1}{2}x + 1$.
• Step 4: Compare with provided choices:

The equation $y = \frac{1}{2}x + 1$ matches choice 4. Therefore, the correct equation corresponding to the table is $y = \frac{1}{2}x + 1$.

To solve this problem, we'll follow these steps:

• Step 1: Determine the form of the potential equation.
• Step 2: Calculate the slope using two points from the table.
• Step 3: Identify the y-intercept.

Now, let's work through each step:

Step 1: Determine the Equation Form
Since the relationship appears linear, we'll use $y = mx + b$.

Step 2: Calculate the Slope
Using the points $(-1, 1)$ and $(0, 2)$, calculate the slope:
$m = \frac{2 - 1}{0 - (-1)} = \frac{1}{1} = 1$.

Step 3: Identify the Y-Intercept
Using the slope $m = 1$ and point $(0, 2)$, since the y-intercept $b$ is the $y$-value when $x = 0$, $b = 2$.

Thus, the equation is $y = x + 2$.

Verify by checking all points from the table:
- For $X = -1, Y = 1$: $1 = -1 + 2$
- For $X = 0, Y = 2$: $2 = 0 + 2$
- For $X = 1, Y = 3$: $3 = 1 + 2$
- For $X = 2, Y = 4$: $4 = 2 + 2$
- For $X = 3, Y = 5$: $5 = 3 + 2$

Thus the equation $y = x + 2$ satisfies all table values.

Therefore, the solution to the problem is $y = x + 2$.