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Answer :
To find the domain and range of the function [tex]f(x) = -\frac{3}{5}x^3[/tex], we need to examine the properties of the function step by step.
Understanding the Function:
The function is a cubic function, which means its form is based on [tex]x^3[/tex]. Cubic functions generally have the form of [tex]ax^3 + bx^2 + cx + d[/tex], and they can take on any real number value as [tex]x[/tex] varies.Finding the Domain:
The domain of a function is the set of all possible input values ([tex]x[/tex]) that the function can accept. For polynomial functions, which include cubic functions like this one, there are no restrictions on the values that [tex]x[/tex] can take. Therefore, the domain is all real numbers:[tex]ext{Domain: } (-\infty, \infty)[/tex]
Finding the Range:
The range of a function is the set of all possible output values ([tex]f(x)[/tex]). Since [tex]f(x) = -\frac{3}{5}x^3[/tex] is a cubic function multiplied by a negative coefficient, it will output negative values as well as positive values depending on the value of [tex]x[/tex]:- As [tex]x[/tex] approaches positive infinity, [tex]f(x)[/tex] approaches negative infinity.
- As [tex]x[/tex] approaches negative infinity, [tex]f(x)[/tex] also approaches positive infinity.
- The function will cross the x-axis at [tex]x = 0[/tex], giving us a value of [tex]0[/tex] as well.
Therefore, the range of this function is:
[tex]ext{Range: } (-\infty, 0) \cup (0, \infty)[/tex]
Based on this analysis, the correct option for the domain and range of the function [tex]f(x) = -\frac{3}{5}x^3[/tex] corresponds to:
Option a)
Domain: [tex](-\infty, \infty)[/tex]
Range: [tex](-\infty, 0)[/tex]
Therefore, the selected answer is option a.
Complete Question:
What are the domain and range of the function?
f(x) = -3/5x^3
a.) Domain: (-∞, ∞)
Range: (-∞, 0)
b.) Domain: (-∞, 0) U (0, ∞)
Range: (-∞, 0)
c.) Domain: (-∞, 0) U (0, ∞)
Range: (-∞, 0) U (0, ∞)
d.) Domain: (-∞, 0) U (0, ∞)
Range: (0, ∞)
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Rewritten by : Barada
The domain and range of the function are:
Domain: (- ∞,0) U (0, ∞)
Range: ( - ∞,0) U (0, ∞)
Domain and Range of function
The domain are the input values for which a function exist while the range are output value for which the function exist.
Given the function:
g(x) = 35x^5
The function can exist on all real values since we will have a correspoding value of y for every value of x.
Hence the domain and range of the function are:
Domain: (- ∞,0) U (0, ∞)
Range: ( - ∞,0) U (0, ∞)
Learn more on domain and range here: https://brainly.com/question/2264373
