Questions: Check the Asymptotes box. Use the a, b, and c-sliders to graph the function f(x)=(x-1)/(x^2+x-2). What is the domain of f(x)?

Transcript text: Check the Asymptotes box. Use the a, $b$, and $c$-sliders to graph the function $f(x)=\frac{x-1}{x^{2}+x-2}$. What is the domain of $f(x)$ ?

Solution

Solution Steps

Step 1: Identify the Asymptotes

The function given is $ f(x) = \frac{x-1}{x^2 + x - 2} $. To find the asymptotes, we need to factor the denominator and identify the values of $ x $ that make the denominator zero.

Step 2: Factor the Denominator

The denominator is $ x^2 + x - 2 $. Factoring this, we get: \[ x^2 + x - 2 = (x - 1)(x + 2) \]

Step 3: Determine the Domain

The domain of $ f(x) $ is all real numbers except where the denominator is zero. From the factored form, the denominator is zero at $ x = 1 $ and $ x = -2 $. Therefore, the domain of $ f(x) $ is all real numbers except $ x = 1 $ and $ x = -2 $.

Final Answer

The domain of $ f(x) = \frac{x-1}{x^2 + x - 2} $ is $ x \in \mathbb{R} \setminus \{1, -2\} $.

{"axisType": 3, "coordSystem": {"xmin": -5, "xmax": 5, "ymin": -5, "ymax": 5}, "commands": ["y = (x-1)/((x-1)*(x+2))"], "latex_expressions": ["$f(x) = \\frac{x-1}{x^2 + x - 2}$"]}

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