To determine the domain and range of the function
[tex]\[ f(x)=-\frac{3}{5x^3}, \][/tex]
we need to analyze both the conditions for the function to be defined and the values it can take.
### Domain
To find the domain, we need to identify the values of [tex]\( x \)[/tex] for which the function is defined. Here, the function involves a division by [tex]\( 5x^3 \)[/tex]. Division by zero is undefined, so we need to ensure that [tex]\( 5x^3 \neq 0 \)[/tex].
1. [tex]\( 5x^3 = 0 \)[/tex] when [tex]\( x = 0 \)[/tex].
Thus, [tex]\( x \)[/tex] cannot be zero. Therefore, the domain includes all real numbers except [tex]\( x = 0 \)[/tex]. This can be expressed as:
[tex]\[ (-\infty, 0) \cup (0, \infty). \][/tex]
### Range
To determine the range, we need to consider the outputs of the function for all inputs within its domain.
[tex]\[ f(x) = -\frac{3}{5x^3} \][/tex]
Let's examine the behavior of the function as [tex]\( x \)[/tex] takes on different values within the domain:
1. When [tex]\( x \)[/tex] is a large positive number, [tex]\( x^3 \)[/tex] is positive and large, making [tex]\( \frac{3}{5x^3} \)[/tex] a small positive number. The negative sign makes [tex]\( f(x) \)[/tex] a small negative number.
2. As [tex]\( x \)[/tex] approaches 0 from either side, [tex]\( \frac{3}{5x^3} \)[/tex] becomes very large in magnitude because the denominator is small. As [tex]\( x \)[/tex] gets close to zero from the positive side, [tex]\( f(x) \)[/tex] becomes a very large negative number.
3. When [tex]\( x \)[/tex] is negative, [tex]\( x^3 \)[/tex] is negative, and [tex]\( \frac{3}{5x^3} \)[/tex] becomes negative. The negative sign in the function makes [tex]\( f(x) \)[/tex] positive, which is not possible since the function is designed to yield only negative values.
Therefore, every [tex]\( f(x) \)[/tex] is a negative number. The function can take any negative value, getting arbitrarily close to zero (but never reaching zero) from the negative side, and reaching negative infinity as [tex]\( x \)[/tex] gets closer to zero.
So, the range of the function is:
[tex]\[ (-\infty, 0). \][/tex]
### Conclusion
Based on this analysis, we conclude that:
- The domain is [tex]\( (-\infty, 0) \cup (0, \infty) \)[/tex].
- The range is [tex]\( (-\infty, 0) \)[/tex].
Therefore, the correct answer is:
[tex]\[ \text{Domain: } (-\infty, 0) \cup (0, \infty) \][/tex]
[tex]\[ \text{Range: } (-\infty, 0) \][/tex]
