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.