Given various values of the linear functions $f(x)$ and $g(x)$ in the table, determine the $y$-intercept of $(f-g)(x)$.
| x | -6 | -4 | -1 | 3 | 4 |
| :---- | :---- | :---- | :---- | :---- | :---- |
| f(x) | -36 | -26 | -11 | 9 | 14 |
| g(x) | 15 | 11 | 5 | -3 | -5 |
A. $(0,9)$
B. $(0,3)$
C. $(0,-3)$
D. $(0,-9)$
Answer (1)
Find the equation of f ( x ) using two points from the table: f ( x ) = 5 x − 6 , so f ( 0 ) = − 6 .
Find the equation of g ( x ) using two points from the table: g ( x ) = − 2 x + 3 , so g ( 0 ) = 3 .
Calculate ( f − g ) ( 0 ) = f ( 0 ) − g ( 0 ) = − 6 − 3 = − 9 .
The y -intercept of ( f − g ) ( x ) is − 9 .
Explanation
Understanding the Problem We are given a table of values for two linear functions, f ( x ) and g ( x ) , and we want to find the y -intercept of the function ( f − g ) ( x ) . The y -intercept is the value of the function when x = 0 , so we need to find ( f − g ) ( 0 ) = f ( 0 ) − g ( 0 ) .
Finding the Equation of f(x) First, we need to find the equations of the linear functions f ( x ) and g ( x ) . We can use two points from the table to find the slope and y -intercept of each function. Let's use the points ( − 6 , − 36 ) and ( − 4 , − 26 ) for f ( x ) . The slope of f ( x ) is given by: m f = − 4 − ( − 6 ) − 26 − ( − 36 ) = 2 10 = 5 Now, we can use the point-slope form of a linear equation, y − y 1 = m ( x − x 1 ) , with the point ( − 6 , − 36 ) :
f ( x ) − ( − 36 ) = 5 ( x − ( − 6 )) f ( x ) + 36 = 5 ( x + 6 ) f ( x ) = 5 x + 30 − 36 f ( x ) = 5 x − 6 So, f ( 0 ) = 5 ( 0 ) − 6 = − 6 .
Finding the Equation of g(x) Next, let's find the equation of g ( x ) using the points ( − 6 , 15 ) and ( − 4 , 11 ) . The slope of g ( x ) is given by: m g = − 4 − ( − 6 ) 11 − 15 = 2 − 4 = − 2 Now, we can use the point-slope form of a linear equation with the point ( − 6 , 15 ) :
g ( x ) − 15 = − 2 ( x − ( − 6 )) g ( x ) − 15 = − 2 ( x + 6 ) g ( x ) = − 2 x − 12 + 15 g ( x ) = − 2 x + 3 So, g ( 0 ) = − 2 ( 0 ) + 3 = 3 .
Calculating (f-g)(0) Now we can find ( f − g ) ( 0 ) = f ( 0 ) − g ( 0 ) :
( f − g ) ( 0 ) = − 6 − 3 = − 9 Therefore, the y -intercept of ( f − g ) ( x ) is − 9 .
Examples
Understanding linear functions and their intercepts is crucial in many real-world applications. For instance, in business, if f ( x ) represents the revenue and g ( x ) represents the cost at x units sold, then ( f − g ) ( x ) represents the profit. The y -intercept of the profit function, ( f − g ) ( 0 ) , tells you the initial profit (or loss) before any units are sold. This could represent initial investments or fixed costs. Knowing this value helps in making informed business decisions.
