Answer to Question #186527 in Linear Algebra for Petra

Question

What are the domain and range for these equations?

1. f(x)= square root of (x-1)
2. f(x)=6x-1/1-2x
3. the inverse of function f(x) = square root of (x-1)

Answer 1

Solution.

1.

$$f(x)=\sqrt{x-1}$$

Domain: $x \in [1,\infty).$

Range: $y\in [0,\infty).$

2.

$$f(x)=\frac{6x-1}{1-2x}$$

Domain: $x \in R \backslash\{\frac{1}{2}\}.$

$\lim_{x\to\pm\infty}\frac{6x-1}{1-2x}=\lim_{x\to\pm\infty}\frac{6-\frac{1}{x}}{\frac{1}{x}-2}=\frac{6}{-2}=-3.$

So,

Range: $y\in (-\infty,\infty)\backslash\{-3\}.$

3.

$$f(x)=\sqrt{x-1},\newline f^2(x)=x-1,\newline x=f^2(x)+1,\newline f^{-1}(x)=x^2+1, x\geq0.$$

Domain: $x\in [0,\infty).$

Range: $y\in [1,\infty).$