Given the following piecewise function
f(x)= {-x+1 for x < 0
{-1 for 0 ≤ x ≤ 3
{-2x for x > 3
a) Find the domain
b) Find the range
c) Find the intercepts
d) Is f continuous on its domain? If not, state where f is discontinuous.
e) Graph the function
the magnitude of dot and products of two vectors are 6 ander root 3 and 6 respectivly9.find the angle between the vectors.
a) To find the domain of the function, we need to identify all the values of x for which the function is defined. In this case, we have three different piecewise definitions.
For the first piece of the function, f(x) = -x + 1, the domain is all x values less than 0.
For the second piece, f(x) = -1, the domain is all x values between 0 and 3 (inclusive).
For the third piece, f(x) = -2x, the domain is all x values greater than 3.
Therefore, the domain for the given piecewise function is the union of all these intervals: (-∞, 0) U [0, 3] U (3, ∞).
b) To find the range of the function, we need to determine all the possible output values, or all the values that f(x) can take on. Looking at the function definitions:
For the first piece, -x + 1, the range will be all values less than or equal to 1.
For the second piece, -1, the range is simply -1.
For the third piece, -2x, the range is all values greater than or equal to -∞.
Thus, the range for the given piecewise function is (-∞, 1] U {-1} U (-∞, ∞), which simplifies to (-∞, 1] U {-1}.
c) To find the x-intercepts, we need to determine the values of x for which f(x) = 0. In this case, we only have one piece of the function where f(x) = 0:
For the third piece, f(x) = -2x, setting this equal to zero gives us -2x = 0. Solving for x, we find x = 0.
Therefore, the function has only one x-intercept, which is x = 0.
To find the y-intercept, we need to determine the value of f(0). According to the second piece of the function, we have f(0) = -1. Thus, the y-intercept is -1.
d) To determine if f is continuous on its domain, we need to check if the function is continuous at the boundaries of each piece.
At x = 0, we have f(0) = -1, which matches the value given by the second piece (-1). Therefore, f is continuous at x = 0.
To determine if there are any discontinuities, we need to check for any "jumps" or abrupt changes within each piece. In this case, there are no jumps in any of the pieces.
Therefore, the function f is continuous on its domain.
e) To graph the given piecewise function, we plot the three pieces of the function separately.
For the first piece, -x + 1, the line will have a negative slope and intersect the y-axis at 1. We draw a dashed line for x < 0.
For the second piece, -1, the line will be a horizontal line with a y-value of -1. We draw a solid line for 0 ≤ x ≤ 3.
For the third piece, -2x, the line will have a negative slope and pass through the origin. We draw another dashed line for x > 3.
By connecting these three portions of the graph, we get the complete graph of the piecewise function.