A function is given such that f(4)−f(6)=9. If the function is linear, which of the following could the function possibly be?(1 point)
A. f(x)=−4.5x+b for some constant b
B. f(x)=−9x+b for some constant b
C. f(x)=9x+b for some constant b
D. f(x)=4.5x+b for some constant b
D. f(x)=4.5x+b for some constant b
why, explain
To determine which of the given functions could possibly be linear, we need to analyze the differences in the expression f(4) - f(6) when using each function.
Let's test each function option:
A. f(x) = -4.5x + b
Using this function, f(4) - f(6) = (-4.5 * 4 + b) - (-4.5 * 6 + b) = (-18 + b) - (-27 + b) = -18 + b + 27 - b = 9.
Therefore, Option A is a possible linear function.
B. f(x) = -9x + b
Using this function, f(4) - f(6) = (-9 * 4 + b) - (-9 * 6 + b) = (-36 + b) - (-54 + b) = -36 + b + 54 - b = 18.
Therefore, Option B is not a possible linear function.
C. f(x) = 9x + b
Using this function, f(4) - f(6) = (9 * 4 + b) - (9 * 6 + b) = (36 + b) - (54 + b) = 36 + b - 54 - b = -18.
Therefore, Option C is not a possible linear function.
D. f(x) = 4.5x + b
Using this function, f(4) - f(6) = (4.5 * 4 + b) - (4.5 * 6 + b) = (18 + b) - (27 + b) = 18 + b - 27 - b = -9.
Therefore, Option D is not a possible linear function.
In conclusion, the only possible linear function is A. f(x) = -4.5x + b for some constant b.
To determine which of the given options is a possible linear function, we need to analyze the given equation f(4) - f(6) = 9.
For a linear function, the difference in the values of f(x) for any two points should be proportional to the difference in the x-values. In other words, if we subtract f(x₁) from f(x₂), the result should be equal to the constant multiple of (x₂ - x₁).
Let's consider the given equation: f(4) - f(6) = 9
The difference in x-values is 6 - 4 = 2.
Now, let's check each option and see if they satisfy this condition:
A. f(x) = -4.5x + b: f(4) - f(6) = (-4.5*4 + b) - (-4.5*6 + b) = (-18 + b) - (-27 + b) = b - b + 27 - 18 = 9
B. f(x) = -9x + b: f(4) - f(6) = (-9*4 + b) - (-9*6 + b) = (-36 + b) - (-54 + b) = b - b + 54 - 36 = 18
C. f(x) = 9x + b: f(4) - f(6) = (9*4 + b) - (9*6 + b) = (36 + b) - (54 + b) = b - b + 54 - 36 = 18
D. f(x) = 4.5x + b: f(4) - f(6) = (4.5*4 + b) - (4.5*6 + b) = (18 + b) - (27 + b) = b - b + 27 - 18 = 9
From the calculations, option A, B, C, and D all result in f(4) - f(6) = 9. Therefore, all options are possible functions that satisfy the given condition.