a function f is defined as f(x)= square root of (x+4).
a) identify the domain of the function
b) identify the range of the function
c)Compute the intercept
d) compute the x intercept
E) sketch a graph of the function
To find the domain of the function f(x) = √(x + 4), we need to determine the set of all possible values for x that make the function defined.
a) Domain: The function f(x) is defined for any real number that, when substituted into the expression (x + 4), does not result in the square root of a negative number. In this case, the expression inside the square root should be greater than or equal to zero.
x + 4 ≥ 0
x ≥ -4
Therefore, the domain of f(x) is x ≥ -4.
b) Range: To find the range of the function, we look for the set of all possible values the function can output.
Since the square root of a number is always greater than or equal to zero, the range of f(x) is f(x) ≥ 0.
c) y-intercept: The y-intercept is the point where the graph of a function intersects the y-axis. In this case, to find the y-intercept, we set x = 0 and calculate f(0):
f(0) = √(0 + 4) = √4 = 2
So, the y-intercept is (0, 2).
d) x-intercept: The x-intercept is the point where the graph of a function intersects the x-axis. To find the x-intercept, we set f(x) = 0 and solve for x:
0 = √(x + 4)
0 = x + 4
x = -4
So, the x-intercept is (-4, 0).
e) Graph: To sketch the graph of the function, start with the y-intercept at (0, 2). Then plot some additional points by substituting different values of x into the function and calculating f(x). Connect the points with a smooth curve, noting that the graph will approach but never touch the x-axis as x approaches -4.