1. Which of the following equations correctly describes the function in the table using function notation?

X: 0, 1, 2, 3
Y: -5, -9, -13, -17
A) f(y) = -5x - 4
B) f(x) = -4x - 5
C) f(x) = -5x - 4
D) f(y) = -4x - 5

2. Which equation describes the function shown in the mapping diagram?
X: 0, 1, 2, 3
Y: -8, -7 1/2, -7, -6 1/2
A) f(x) = 1/2x - 8
B) f(y) = 1/2x - 8
C) f(y) = 8x - 1/2
D) f(x) = 8x - 1/2

3. Evaluate the function for the following input: f(x) = 3 − 2x.
Which input generates the largest output when evaluating the function?
A) f(2)
B) f(0)
C) f(3)
D) f(5)

User Icon for Bot GPT 4-o mini Bot GPT 4-o mini answered
1 hour ago

Let's address each of your questions one by one.

1. Function Describing Equation

We need to find the equation that correctly represents the values given in the table:

  • For \( x = 0 \), \( y = -5 \)
  • For \( x = 1 \), \( y = -9 \)
  • For \( x = 2 \), \( y = -13 \)
  • For \( x = 3 \), \( y = -17 \)

First, we can calculate the slope (change in \( y \) over change in \( x \)): From points (0, -5) to (1, -9):

  • \( \Delta y = -9 - (-5) = -4 \)
  • \( \Delta x = 1 - 0 = 1 \)

The slope \( m = -4 \).

Now, we can use the point-slope form or direct substitution to find the function: Using \( f(x) = mx + b \):

  • When \( x = 0 \), \( f(0) = -5 \) means \( b = -5 \).
  • So, we have:

\[ f(x) = -4x - 5 \]

Which matches option B: B) f(x) = -4x - 5.


2. Mapping Diagram Function

From the mapping diagram, we have the output values:

  • For \( x = 0 \), \( y = -8 \)
  • For \( x = 1 \), \( y = -7.5 \)
  • For \( x = 2 \), \( y = -7 \)
  • For \( x = 3 \), \( y = -6.5 \)

These points indicate a linear relationship. We can find the slope: From (0, -8) to (1, -7.5):

  • \( \Delta y = -7.5 - (-8) = 0.5 \)
  • \( \Delta x = 1 - 0 = 1 \)

The slope \( m = 0.5 \). Using \( y = mx + b \):

  • When \( x = 0 \), \( y = -8 \), so \( b = -8 \).

The function is: \[ f(x) = \frac{1}{2}x - 8 \]

Which corresponds to option A: A) f(x) = \frac{1}{2}x - 8.


3. Evaluate the Function

To determine which input generates the largest output for \( f(x) = 3 - 2x \), we will evaluate the function at each option:

  • \( f(0) = 3 - 2(0) = 3 \)
  • \( f(2) = 3 - 2(2) = 3 - 4 = -1 \)
  • \( f(3) = 3 - 2(3) = 3 - 6 = -3 \)
  • \( f(5) = 3 - 2(5) = 3 - 10 = -7 \)

The values of the function are:

  • \( f(0) = 3 \)
  • \( f(2) = -1 \)
  • \( f(3) = -3 \)
  • \( f(5) = -7 \)

The largest output is from \( f(0) \): B) f(0) generates the largest output.