The distance between the y-intercept of the function f(x)=3(2)ˣ and the y-intercept of the function g(x) for this case is given by: Option C: 8 units
What is y-intercept of a function?
The intersection of the graph of the function with the y-axis gives y-intercept of that function. The y-intercept is the value of y on the y-axis at which the considered function intersects it.
Assume that we've got: y = f(x)
At y-axis, we've got x = 0, so putting it will give us the y-intercept.
Thus, y-intercept of y = f(x) is y = f(0)
What is the equation of a line passing through two given points in 2 dimensional plane?
Suppose the given points are [tex](x_1, y_1)[/tex] and [tex](x_2, y_2)[/tex], then the equation of the straight line joining both two points is given by
[tex](y - y_1) = \dfrac{y_2 - y_1}{x_2 - x_1} (x -x_1)[/tex]
We're given that:
[tex]f(x) = 3(2)^x[/tex]
Its y-intercept is:
[tex]y = f(0) = 3(2)^0 = 3\times 1 = 3[/tex]
Assuming y = g(x) is a linear function, from the first two points [tex](x_1, y_1) = (-5,15)[/tex] and [tex](x_2, y_2) = (-2, 3)[/tex], we get:
[tex](y - 15) = \dfrac{3 - 15}{-2 - (-5)} (x -(-5))\\\\y -15 = -\dfrac{12}{3} (x+5)\\\\y-15 = -4x - 20\\y = -4x - 5[/tex]
Checking if really y = g(x) is linear or not by evaluating it on its rest of the points:
[tex]y = g(2) = -4(2) -5 =-8-5 = -13\\\\y = g(5) = -4(5) -5 = -25[/tex]
These outputs matches with the table output, so we can take:
[tex]g(x) = -4x -5[/tex]
Its y-intercept is:
[tex]y = g(0) = -4(0) -5 = -5[/tex]
The distance between 3 and -5 on the y-axis is the absolute difference between these values of y, evaluated as:
[tex]d = |3 - (-5)| = |8| = 8[/tex] units.
Thus, the distance between the y-intercept of the function f(x)=3(2)ˣ and the y-intercept of the function g(x) for this case is given by: Option C: 8 units.
Learn more about straight line equation here:
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