The equations for two functions are shown below.
f(x) = x + 3 and g(x) = 2x - 5
Which intercept is farthest from the origin compared to the other three intercepts?
A) x-intercept of f(x)
B) y-intercept of f(x)
C) x-intercept of g(x)
D) y-intercept of g(x)
f(x) x:-3, y:3
g(x) x:5/2, y:-5
so, which is farthest from (0,0)?
To determine which intercept is farthest from the origin, we need to find the distance between each intercept and the origin.
The x-intercept of a function is the point where the graph intersects the x-axis, so for f(x), we set y = 0:
0 = x + 3
x = -3
Therefore, the x-intercept of f(x) is -3.
Similarly, the x-intercept of g(x) is found by setting y = 0:
0 = 2x - 5
x = 5/2 or 2.5
So, the x-intercept of g(x) is 2.5.
The y-intercept of a function is the point where the graph intersects the y-axis when x = 0.
For f(x), when x = 0:
f(0) = 0 + 3
f(0) = 3
Thus, the y-intercept of f(x) is 3.
For g(x), when x = 0:
g(0) = 2(0) - 5
g(0) = -5
Hence, the y-intercept of g(x) is -5.
Now, let's calculate the distances of each intercept from the origin:
For the x-intercept of f(x) (-3), the distance from the origin is |-3 - 0| = 3.
For the y-intercept of f(x) (3), the distance from the origin is |0 - 3| = 3.
For the x-intercept of g(x) (2.5), the distance from the origin is |2.5 - 0| = 2.5.
For the y-intercept of g(x) (-5), the distance from the origin is |0 - (-5)| = 5.
Comparing these distances, the farthest intercept from the origin is the y-intercept of g(x) which is -5.
Therefore, the correct answer is the D) y-intercept of g(x).
To determine which intercept is farthest from the origin compared to the other three intercepts, we need to find the distances of each intercept from the origin. Let's find the coordinates of the intercepts for the given functions:
For f(x) = x + 3:
- The x-intercept is found by setting y = 0 and solving for x. In this case, x + 3 = 0, so x = -3. The x-intercept has coordinates (-3, 0).
- The y-intercept is found by setting x = 0 and solving for y. In this case, y = 0 + 3, so y = 3. The y-intercept has coordinates (0, 3).
For g(x) = 2x - 5:
- The x-intercept is found by setting y = 0 and solving for x. In this case, 2x - 5 = 0, so x = 5/2. The x-intercept has coordinates (5/2, 0).
- The y-intercept is found by setting x = 0 and solving for y. In this case, y = 2(0) - 5, so y = -5. The y-intercept has coordinates (0, -5).
Now we calculate the distances of each intercept from the origin using the distance formula, which is given by the equation sqrt(x^2 + y^2). Let's calculate the distances:
For the x-intercept of f(x): sqrt((-3)^2 + 0^2) = sqrt(9 + 0) = sqrt(9) = 3
For the y-intercept of f(x): sqrt(0^2 + 3^2) = sqrt(0 + 9) = sqrt(9) = 3
For the x-intercept of g(x): sqrt((5/2)^2 + 0^2) = sqrt(25/4 + 0) = sqrt(25/4) = 5/2
For the y-intercept of g(x): sqrt(0^2 + (-5)^2) = sqrt(0 + 25) = sqrt(25) = 5
Comparing the distances, we can see that the farthest intercept from the origin is the y-intercept of g(x) with a distance of 5 units. Therefore, the answer is option D) y-intercept of g(x).