the table below shows some values for the function f. if f is a linear function, what is the value of a+b?
table:
X | f(x)
0 | a
1 | 12
2 | b
since the function is linear, the slope must be constant for any pair
so (12-a)/(1-0) = (b-12)/(2-1)
12-a = b-12
24 = a+b
Well, well, well... looks like we have a little mystery to solve here. Let's put on our detective hats and investigate.
We know that if f is a linear function, it can be represented by a straight line equation of the form f(x) = mx + c, where m is the slope and c is the y-intercept.
Looking at our table, we have f(0) = a, f(1) = 12, and f(2) = b. Hmmm, interesting.
The slope of a linear function can be found by taking the difference in y-coordinates divided by the difference in x-coordinates. In this case, we can see that the difference in x-coordinates is 2 - 1 = 1, and the difference in y-coordinates is b - 12.
So, the slope (m) would be (b - 12)/1 = b - 12.
Since f(0) = a, we know that the y-intercept (c) would be a. Therefore, our equation becomes f(x) = (b - 12)x + a.
Now, to find the value of a + b, we need to solve for x = 0:
f(0) = (b - 12)(0) + a
a = a
Oh, bummer! No matter what the value of b is, it doesn't matter for the calculation of a + b. The mystery remains unsolved, my friend. But hey, every good detective needs some unsolved cases to keep things interesting, right?
To determine the value of a+b, we need to find the values of a and b.
Since f is a linear function, we can use the formula for the slope-intercept form of a linear equation, which is:
f(x) = mx + b
where m is the slope and b is the y-intercept.
Using the given information from the table:
Let's find the slope of the function:
m = (f(x2) - f(x1))/(x2 - x1)
m = (12 - a)/(1 - 0)
m = (12 - a)/1
Since the function is linear, the slope should be constant. Therefore, the slope between any two points on the table should be equal.
So we can calculate the slope between the points (0, a) and (1, 12) as well as between the points (1, 12) and (2, b).
(m1) Slope between (0, a) and (1, 12) = (12 - a)/(1 - 0) = 12 - a
(m2) Slope between (1, 12) and (2, b) = (b - 12)/(2 - 1) = b - 12
Since the slopes are equal, we can set them equal to each other:
12 - a = b - 12
To solve for a+b, we need to solve this equation for a or b. Let's solve for a:
12 - a + a = b - 12 + a
12 = b + a - 12
24 = b + a
Therefore, the value of a+b is 24.
To determine the value of a+b, we need to find the values of a and b.
Given that f is a linear function, we can write its equation as f(x) = mx + c, where m is the slope and c is the y-intercept.
Using the table, we can determine the values of a and b by matching the x-values in the table with the corresponding f(x) values.
From the table, we have f(0) = a, f(1) = 12, and f(2) = b.
Since f(0) = a, we can substitute it into the equation:
a = f(0)
Looking at the table, we see that f(0) corresponds to the value of a. Therefore, a is equal to the value in the 'f(x)' column for x = 0.
Similarly, for f(2) = b, we can substitute it into the equation:
b = f(2)
Looking at the table, we see that f(2) corresponds to the value of b. Therefore, b is equal to the value in the 'f(x)' column for x = 2.
Once we have the values of a and b, we can simply add them together to find the value of a+b.