Range of a function using graph

Question

If we are given a function

$$y = \frac{x-1}{x+2}$$

and we have to find the range .

So I tried, I wrote $$y = 1 - \frac3{x+2}$$ $$y-1 = \frac{-3}{x-2}$$ We know this is an equation of rectangular hyperbola , but how can I find range using its graph?

Answer 1

The function $f(x) = 1/x$ has domain $D_f = (-\infty, 0) \cup (0, \infty)$ and range $R_f = (-\infty, 0) \cup (0, \infty)$.

Based on your work, we can see that the graph of the function $$g(x) = \frac{x - 1}{x + 2} = 1 - \frac{3}{x + 2}$$ is obtained from the graph of $f(x)$ by reflecting the graph of $f(x)$ in the $x$-axis, stretching it vertically by a factor of $3$, and translating it two units to the left and one unit up. Consequently, the domain of $g(x)$ is $D_g = (-\infty, -2) \cup (-2, \infty)$ and its range is $R_g = (-\infty, 1) \cup (1, \infty)$.

Answer 2

Hint. Try taking the limits at $-\infty,~+\infty$ and also the right and left limits at $x_0=-2.$ You can verify that the range is $\mathbb{R}\setminus \{1\}$.

Answer 3

If you can graph it, you can determine its asymptotes. The range will be the set of y values that the curve sweeps over.

Answer 4

As an appendix to Nikolaos Skout's hint, the function looks like below. You should be able to imagine something like this using the usual methods and tricks in calculus for graphing functions.

Answer 5

At $y = 1 - somehyperbola$ you're almost done.

The hyperbola can become anything except $0$. Add or subtract from $1$, you can have anything except $1$.