Question

Use the table to answer the question.
x y
0 3
1 3 1/2
2 4
3 4 1/2

Which of the following equations describes the function in the table using function notation?
Responses:

f(x)=3x+1/2

f(y)=12x+3

f(x)=12x+3

f(y)=3x+1/2

Answer 1

To identify the correct equation that describes the function given in the table, we first need to examine the x and y values provided:

- When \( x = 0 \), \( y = 3 \)
- When \( x = 1 \), \( y = 3.5 \)
- When \( x = 2 \), \( y = 4 \)
- When \( x = 3 \), \( y = 4.5 \)

From this data, we can observe the following changes in \( y \) with respect to \( x \):

- From \( x = 0 \) to \( x = 1 \) (increase of 1): \( y \) increases from 3 to 3.5 (an increase of 0.5)
- From \( x = 1 \) to \( x = 2 \) (increase of 1): \( y \) increases from 3.5 to 4 (an increase of 0.5)
- From \( x = 2 \) to \( x = 3 \) (increase of 1): \( y \) increases from 4 to 4.5 (an increase of 0.5)

This indicates a linear relationship where \( y \) increases by 0.5 for each increase of 1 in \( x \).

To express this relationship in slope-intercept form (which is \( y = mx + b \)), we note:

1. The slope \( m \) is 0.5 (since \( 4.5 - 3 = 1.5 \) over the change in \( x = 3 - 0 = 3 \), gives a slope of \( \frac{1.5}{3} = 0.5 \)).
2. The y-intercept \( b \) is 3 (when \( x = 0 \)).

Thus, the equation can be derived as:
\[ y = 0.5x + 3 \]
or, converting the slope to a fraction,
\[ y = \frac{1}{2} x + 3 \]

Using function notation, this can be written as:
\[ f(x) = \frac{1}{2}x + 3 \]

However, simplified further, it can also be expressed as:
\[ f(x) = \frac{1}{2} x + 3 \]

Now inspecting the given responses, we confirm:
- **f(x)=3x+1/2** (incorrect)
- **f(y)=12x+3** (incorrect)
- **f(x)=12x+3** (incorrect)
- **f(y)=3x+1/2** (incorrect)

Considering that there might be a typographical error in the options presented and what we derived matches none exactly, **none of the responses appear correct** based on the provided data.

If there should have been an option such as:
\[ f(x) = \frac{1}{2} x + 3 \]
or similar with correct formatting, it would have been correct. Make sure to recheck the function choices given or clarify if there was an input error.