Answer:
- domain: {x ∈ ℝ : x ≤ 5}
- range: {y ∈ ℝ : y ≤ -1}
Explanation:
Domain
The domain of a function is the set of x values for which the function is defined. Here, the domain is limited by the values of x that make the square root defined. That is, the expression under the radical cannot be negative:
-3x +15 ≥ 0
15 ≥ 3x . . . . . . add 3x
5 ≥ x . . . . . . . . divide by 3
x ≤ 5 . . . . . . . . put x on the left (swap sides)
The rest of the notation in the domain expression simply says x is a real number.
domain: {x ∈ ℝ : x ≤ 5} . . . . . . matches the first choice
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Range
The range of a function is the set of values that f(x) can have. We know the square root can be zero or any positive number. When it is zero, f(x) = -1.
When it is a positive number, that value is multiplied by -4 and added to -1, so f(x) is a number more negative than -1. Then the range of the function is all numbers -1 and below:
range: {y ∈ ℝ : y ≤ -1} . . . . . . matches the last choice
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Comment on domain/range problems
When working domain and range problems, it works well to have a good understanding of the domain and range limitations of the functions we usually work with: polynomials, square root, logarithm, trig functions, exponential functions. Domain and range problems generally involve combinations of these or ratios of combinations of these.