2. describe the key features of the function ( f(x)=x^{-\frac{3}{2}} ).
2. describe the key features of the function ( f(x)=x^{-\frac{3}{2}} ).
Answer
Explanation:
Step1: Domain
For (F(x)=x^{-\frac{3}{2}}=\frac{1}{x^{\frac{3}{2}}}=\frac{1}{\sqrt{x^{3}}}), the expression under the square - root (x^{3}>0) (since the denominator cannot be zero). Solving (x^{3}>0) gives (x > 0). So the domain is ((0,\infty)).
Step2: Range
As (x>0), when (x\to0^{+}), (F(x)=x^{-\frac{3}{2}}=\frac{1}{x^{\frac{3}{2}}}\to\infty). When (x\to\infty), (F(x)=x^{-\frac{3}{2}}=\frac{1}{x^{\frac{3}{2}}}\to0). So the range is ((0,\infty)).
Step3: Continuity
The function (y = x^{-\frac{3}{2}}) is a power function. Since (x>0), and using the formula for power - functions (y = x^{n}) (where (n=-\frac{3}{2})), and the rules of continuity for functions of the form (y=\frac{1}{\sqrt{x^{3}}}) (a composition of a root function and a rational function), it is continuous on its domain ((0,\infty)).
Step4: Asymptotes
- Vertical Asymptote: As (x\to0^{+}), (y = x^{-\frac{3}{2}}\to\infty). So (x = 0) (the (y) - axis) is a vertical asymptote.
- Horizontal Asymptote: As (x\to\infty), (y=x^{-\frac{3}{2}}=\frac{1}{x^{\frac{3}{2}}}\to0). So (y = 0) (the (x) - axis) is a horizontal asymptote.
Step5: Monotonicity
Find the derivative using the power rule (y=x^{n}), (y^\prime=nx^{n - 1}). For (y=x^{-\frac{3}{2}}), (y^\prime=-\frac{3}{2}x^{-\frac{3}{2}-1}=-\frac{3}{2}x^{-\frac{5}{2}}=-\frac{3}{2x^{\frac{5}{2}}}). Since (x>0), (y^\prime<0) for all (x\in(0,\infty)). So the function is decreasing on the interval ((0,\infty)).
Answer:
- Domain: ((0,\infty))
- Range: ((0,\infty))
- Continuity: Continuous on ((0,\infty))
- Vertical Asymptote: (x = 0)
- Horizontal Asymptote: (y = 0)
- Monotonicity: Decreasing on ((0,\infty))