Suppose that the functions $\newline$ $f$ and $\newline$ $g$ are defined as follows. $\newline$ $f(x)=x-1\quad g(x)=\frac{x-1}{x-6}$ $\newline$ Find $\newline$ $\frac{f}{g}$ . Then, give its domain using an interval or union of intervals. $\newline$ Simplify your answers. $\newline$ $\left(\frac{f}{g}\right)(x)=\left[\right.$ $\newline$ Domain of $\newline$ $\frac{f}{g}$ : $\square$ $\newline$ \(\left[\left(\frac{\square}{\square}\right),\square^{\square},\left(\square,\square\right)\right],\left[\left[\square,\square\right],\square\cup\square,\left(\square,\square\right]\right],\left[\left[\square,\square\right),\varnothing,\infty\right],\left[-\infty,\square\right],\left[x\neq\square\right]:\}

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Simplify radical expressions with variables

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Q. Suppose that the functions

\newline

f

and

\newline

g

are defined as follows.

\newline

f(x)=x-1\quad g(x)=\frac{x-1}{x-6}

\newline

Find

\newline

\frac{f}{g}

. Then, give its domain using an interval or union of intervals.

\newline

Simplify your answers.

\newline

\left(\frac{f}{g}\right)(x)=\left[\right.

\newline

Domain of

\newline

\frac{f}{g}

\square

\newline

\(\left[\left(\frac{\square}{\square}\right),\square^{\square},\left(\square,\square\right)\right],\left[\left[\square,\square\right],\square\cup\square,\left(\square,\square\right]\right],\left[\left[\square,\square\right),\varnothing,\infty\right],\left[-\infty,\square\right],\left[x\neq\square\right]:\}

Write Given Functions: Write down the given functions and the expression for $(f/g)(x)$ . $\newline$ We have $f(x) = x - 1$ and $g(x) = \frac{x - 1}{x - 6}$ . To find $(f/g)(x)$ , we need to divide $f(x)$ by $g(x)$ . $\newline$ $(f/g)(x) = \frac{f(x)}{g(x)} = \frac{x - 1}{\left(\frac{x - 1}{x - 6}\right)}.$
Simplify Expression: Simplify the expression for $(f/g)(x)$ . To divide by a fraction, we multiply by its reciprocal. So, we have: $(f/g)(x) = (x - 1) \times \left(\frac{x - 6}{x - 1}\right)$ .
Cancel Common Factors: Cancel out the common factors in the numerator and the denominator. $\newline$ The $(x - 1)$ terms cancel out, leaving us with: $\newline$ $(\frac{f}{g})(x) = x - 6$ .
Determine Domain: Determine the domain of $(f/g)(x)$ . The domain of $(f/g)(x)$ is all real numbers except where $g(x)$ is undefined. Since $g(x)$ is undefined when the denominator is zero, we set $x - 6 = 0$ and solve for $x$ . $x - 6 = 0$ $x = 6$ So, the domain of $(f/g)(x)$ is all real numbers except $x = 6$ .
Write Domain in Interval Notation: Write the domain in interval notation. $\newline$ The domain of $(f/g)(x)$ is all real numbers except $6$ , which can be written as: $\newline$ Domain of $(f/g)$ : $(-\infty, 6) \cup (6, \infty)$ .