Answer :
The answer is:
(Most likely because it’s for 10 something)
(-47, -7)
The range of the function f(x) = -5|x+2| -7, over the interval [10, -5] is [-47, -7]. Hence, option B is the right choice.
What is a function?
A function is a relation between a dependent variable f(x) and an independent variable x, where for every value of x in its domain, there is one and only one value of f(x).
What is the domain of a function?
The domain of a function is the set of values of x, for which f(x) exists.
What is the range of a function?
The range of a function is the set of values of f(x), in a given domain of x.
How do we solve the given question?
We are given a function,
[tex]f(x) = -5\left | x+2 \right | -7[/tex].
We can write this function as,
[tex]f(x)= -5(x+2)-7[/tex] , for x ≥ -2, and
[tex]f(x) = 5(x+2)-7[/tex] , for x ≤ -2.
To calculate the range, we let y = f(x).
We need to calculate the range of f(x) over the interval [-10, 5].
For x ≤ -2,
y = 5(x + 2) -7
or, y = 5x + 10 - 7 = 5x + 3
or, x = (y - 3)/5.
This is a continuous function, so y is defined at all points between (-10 and -2)
The value of y at x = -10 is,
y = 5(-10 + 2) -7 = 5*(-8) - 7 = -40 - 7 = -47.
The value of y at x = -2 is,
y = 5(-2 + 2) -7 = 5*(0) - 7 = -7.
∴ Range in the interval [-10, -2] is [-47, -7].
For x ≥ -2,
y = -5(x + 2) -7
or, y = -5x - 10 - 7 = -5x -17
or, x = -(y + 17)/5.
This is a continuous function, so y is defined at all points between (-2 and 5)
The value of y at x = -2 is,
y = -5(-2 + 2) -7 = -5*(0) - 7 = -7.
The value of y at x = -2 is,
y = -5(5 + 2) -7 = -5*(7) - 7 = -35 - 7 = -42
∴ Range in the interval [-2, 5] is [-42, -7].
∴ Range in the interval [-10, 5] = Range in the interval [-10, -2] ∪ Range in the interval [-2, 5] = [-47, -7] ∪ [-42, -7] = [-47, -7].
∴ The range of the function f(x) = -5|x+2| -7, over the interval [10, -5] is [-47, -7]. Hence, option B is the right choice.
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