Question

two employees each commute from their home to work using various methods of transportation. the function f(x) represents the distance, in miles, that employee 1 is from work x minutes after the start of the commute. the function g(x) represents the distance, in miles, that employee 2 is from work x minutes after the start of the commute.
which statement is true?
employee 2 is 20 miles closer to work than employee 1 at the start of the commute.
employee 2 is 5 miles closer to work than employee 1 at the start of the commute.
employee 1 is 20 miles closer to work than employee 2 at the start of the commute.
employee 1 is 5 miles closer to work than employee 2 at the start of the commute.

Explanation:

Step1: Find Employee 1's initial distance

For Employee 1, the graph (distance - time) starts at \( x = 0 \) (start of commute) with \( y = 50 \) miles? Wait, no, looking at the graph: the x - axis is time (minutes), y - axis is distance (miles). Wait, the graph has a line starting at (0, 50)? Wait, no, the grid: when time \( x = 0 \) (start), the distance for Employee 1: looking at the graph, the y - axis (distance) at \( x = 0 \) (time 0 minutes) – wait, the graph's x is time (minutes), y is distance (miles). Wait, the line starts at (45, 0)? No, maybe I misread. Wait, the table for Employee 2: when \( x = 0 \) (start), \( g(0)=30 \) miles (from the table: \( x = 0 \), \( g(x)=30 \)). For Employee 1, the graph: at time \( x = 0 \) (start of commute), what's the distance? Wait, the graph's y - axis is distance (miles), x - axis is time (minutes). The line goes from (45, 0) to (0, 50)? Wait, no, the coordinates: let's check the graph. The x - axis (time) has 0 at the bottom, 50 at the top? Wait, no, the time is on the y - axis? Wait, the graph is labeled: Time (in minutes) on the y - axis (vertical), Distance (in miles) on the x - axis (horizontal). So the line starts at (50, 45)? No, the point at the bottom: when time \( y = 45 \) minutes, distance \( x = 0 \)? Wait, no, the graph has a line from (0, 45) to (50, 0)? Wait, no, the labels: "Distance (in miles)" on the x - axis (horizontal), "Time (in minutes)" on the y - axis (vertical). So the line starts at (50, 45) and goes to (0, 0)? No, the point at the bottom: when distance \( x = 0 \) (arrived), time \( y = 45 \) minutes? No, this is confusing. Wait, the key is: at the start of the commute (\( x = 0 \)), for Employee 1, what's the distance to work? Wait, the function \( f(x) \) for Employee 1: at \( x = 0 \) (start), the distance is the initial distance. From the graph, when \( x = 0 \) (time 0 minutes), the distance (x - axis) is 50 miles? Wait, no, let's re - examine:

Wait, the table for Employee 2: \( x \) (minutes after start) and \( g(x) \) (distance in miles). When \( x = 0 \) (start), \( g(0)=30 \) miles (distance to work at start). For Employee 1, the graph: when \( x = 0 \) (start of commute), the distance (y - axis? No, x - axis is distance). Wait, the graph's x is distance (miles), y is time (minutes). So the line shows the time taken to cover distance. So at \( x = 50 \) miles (distance to work), time \( y = 0 \) minutes? No, the line starts at \( x = 50 \) miles (distance) with \( y = 45 \) minutes? No, the point at the bottom: when distance \( x = 0 \) (arrived), time \( y = 45 \) minutes. So the initial distance (start of commute, \( x = 50 \) miles) takes 45 minutes? Wait, no, the start of commute is when \( x = 50 \) miles (distance to work) and \( y = 0 \) minutes? No, I think I messed up the axes. Let's use the table for Employee 2: at \( x = 0 \) (start), \( g(0)=30 \) miles (distance to work). For Employee 1, the graph: at \( x = 0 \) (start of commute), the distance to work is 50 miles? Wait, no, the graph's x - axis is distance (miles), so when \( x = 50 \) miles, time \( y = 0 \) minutes (start), and when \( x = 0 \) miles (arrived), time \( y = 45 \) minutes. So Employee 1's initial distance (start of commute) is 50 miles. Employee 2's initial distance (from table, \( x = 0 \)) is \( g(0)=30 \) miles? Wait, no, the table for Employee 2: \( x \) is minutes after start, \( g(x) \) is distance remaining? Wait, yes! \( g(x) \) represents the distance remaining (in miles) after \( x \) minutes. So at \( x = 0 \) (start of commute), \( g(0)=30 \) miles (distance to work). For Employee 1, \( f(x) \) is distance remaining? The graph: at \( x = 0 \) (start), what's \( f(0) \)? The graph: when \( x = 0 \) (start), the distance remaining is 50 miles? Wait, the graph's x - axis is distance (miles), y - axis is time (minutes). So the line from (50, 0) to (0, 45): at \( x = 50 \) miles (start, distance to work), time \( y = 0 \) minutes; at \( x = 0 \) miles (arrived), time \( y = 45 \) minutes. So \( f(0)=50 \) miles (distance to work at start), \( g(0)=30 \) miles (distance to work at start). Wait, no, that can't be. Wait, the question is "closer to work at the start of the commute" – so smaller distance to work at \( x = 0 \).

Wait, Employee 2: \( g(0)=30 \) miles (distance to work at start). Employee 1: from the graph, at \( x = 0 \) (start), the distance to work is 50 miles? Wait, no, maybe the graph is \( f(x) \) as distance remaining, so when \( x = 0 \) (start), \( f(0)=50 \) miles; when \( x = 50 \) miles (distance), time \( y = 45 \) minutes. Wait, no, let's calculate the difference: \( |50 - 30| = 20 \)? No, wait, maybe I had it reversed. Wait, the table for Employee 2: when \( x = 0 \) (start), \( g(0)=30 \); when \( x = 40 \), \( g(40)=0 \) (arrived). For Employee 1, the graph: when \( x = 0 \) (start), distance remaining is 50 miles? When \( x = 50 \) miles (distance), time \( y = 45 \) minutes? No, the y - axis is time, x - axis is distance. So the line goes from (50, 0) to (0, 45), meaning at start (\( x = 50 \) miles, \( y = 0 \) minutes), and arrives at (\( x = 0 \) miles, \( y = 45 \) minutes). So Employee 1's initial distance (start) is 50 miles, Employee 2's initial distance (start) is 30 miles? No, that can't be, because the options are about who is closer. Wait, the options: "Employee 1 is 5 miles closer...", "Employee 2 is 5 miles closer...", "20 miles". Wait, let's re - check the table for Employee 2:

Employee 2's table:
\( x \) (minutes) | \( g(x) \) (distance, miles)
--- | ---
0 | 30
4 | 29
8 | 28
12 | 27
20 | 18
30 | 9
40 | 0

So at \( x = 0 \) (start), \( g(0)=30 \) miles (distance to work).

For Employee 1, the graph: let's find \( f(0) \) (distance to work at start). The graph is a line with distance (x - axis) and time (y - axis). The line passes through (50, 45) and (0, 0)? No, the point at \( x = 50 \) miles (distance) has \( y = 45 \) minutes (time)? No, the graph's x - axis is distance (miles) from 0 to 50, y - axis is time (minutes) from 0 to 50. The line starts at (50, 45) and ends at (0, 0). So at \( x = 0 \) (start), distance is 50 miles? No, \( x = 50 \) is the start distance? Wait, no, when \( x = 0 \) (distance 0 miles, arrived), time \( y = 45 \) minutes. So the start of commute is when distance is 50 miles (x = 50) and time is 0 minutes (y = 0). So \( f(0)=50 \) miles (distance to work at start).

Now, compare \( f(0) \) and \( g(0) \):

\( f(0)=50 \) miles, \( g(0)=30 \) miles? No, that would mean Employee 2 is closer. Wait, the options:

Option 1: Employee 2 is 20 miles closer to work than Employee 1 at the start of the commute.

Wait, \( |50 - 30| = 20 \). Wait, but maybe I got \( f(0) \) wrong. Wait, the graph: maybe the y - axis is distance, x - axis is time. Let's assume the graph is time (x - axis, minutes) and distance (y - axis, miles). So the line starts at (0, 50) and ends at (45, 0). So at \( x = 0 \) (start), \( f(0)=50 \) miles (distance to work). At \( x = 45 \) minutes, \( f(45)=0 \) miles (arrived).

Employee 2: at \( x = 0 \), \( g(0)=30 \) miles (from table).

So the difference in initial distances: \( 50 - 30 = 20 \) miles. So Employee 2 is 20 miles closer? No, wait, \( 50 - 30 = 20 \), so Employee 2's distance (30) is 20 less than Employee 1's (50). So Employee 2 is 20 miles closer to work than Employee 1 at the start of the commute? Wait, no, \( 50 - 30 = 20 \), so Employee 2 is 20 miles closer? Wait, the options:

Option 1: Employee 2 is 20 miles closer to work than Employee 1 at the start of the commute.

Wait, let's confirm:

Employee 1's initial distance (start): from graph, at \( x = 0 \) (time 0), distance to work is 50 miles (since the line goes from (0, 50) to (45, 0) – time 45 minutes to travel 50 miles? No, time on x - axis, distance on y - axis: at \( x = 0 \) minutes, \( y = 50 \) miles (distance to work); at \( x = 45 \) minutes, \( y = 0 \) miles (arrived).

Employee 2's initial distance (start): from table, at \( x = 0 \) minutes, \( g(0)=30 \) miles (distance to work).

Difference: \( 50 - 30 = 20 \) miles. So Employee 2 is 20 miles closer to work than Employee 1 at the start of the commute. Wait, but the options:

First option: Employee 2 is 20 miles closer to work than Employee 1 at the start of the commute.

Yes, that matches. Wait, the options are:

1. Employee 2 is 20 miles closer to work than Employee 1 at the start of the commute.

2. Employee 2 is 5 miles closer...

3. Employee 1 is 20 miles closer...

4. Employee 1 is 5 miles closer...

So the correct option is the first one: "Employee 2 is 20 miles closer to work than Employee 1 at the start of the commute."

Step2: Verify the difference

Employee 1's start distance: \( f(0)=50 \) miles (from graph: at time 0, distance to work is 50).

Employee 2's start distance: \( g(0)=30 \) miles (from table: at time 0, distance to work is 30).

Difference: \( 50 - 30 = 20 \) miles. So Employee 2 is 20 miles closer.

Answer:

Employee 2 is 20 miles closer to work than Employee 1 at the start of the commute. (The option corresponding to this statement)