In Exercises 95–98, use long division to rewrite the equation for g in the form quotient + remainder/divisor. Then use this form of the function's equation and transformations of f(x) = 1/x to graph g. g(x)=(3x−7)/(x−2)

Verified step by step guidance

1. Identify the dividend and divisor for the long division: the dividend is the numerator polynomial $\mathrm{3x\; -\; 7}$ and the divisor is the denominator polynomial $\mathrm{x\; -\; 2}$.
2. Set up the long division by dividing the leading term of the dividend $3x$ by the leading term of the divisor $x$, which gives the first term of the quotient: $3$.
3. Multiply the entire divisor $(x\; -\; 2)$ by the quotient term $3$ to get $3x\; -\; 6$, then subtract this from the dividend $3x\; -\; 7$ to find the remainder.
4. Calculate the remainder by subtracting: $(3x\; -\; 7)\; -\; (3x\; -\; 6)\; =\; -1$. So the remainder is $-1$.
5. Express the function $g(x)$ in the form quotient + remainder/divisor: $g(x)\; =\; 3\; +\; \backslash frac\{-1\}\{x\; -\; 2\}$. Use this form to analyze and graph the function by considering transformations of $f(x)\; =\; \backslash frac\{1\}\{x\}$.

Key Concepts

Polynomial Long Division

Polynomial long division is a method used to divide one polynomial by another, similar to numerical long division. It helps rewrite a rational function as a quotient plus a remainder over the divisor, simplifying the expression for analysis or graphing.

Rational Functions and Their Graphs

Rational functions are ratios of polynomials and often have vertical and horizontal asymptotes. Understanding their behavior, including discontinuities and end behavior, is essential for graphing and interpreting transformations.

Transformations of the Parent Function f(x) = 1/x

The function f(x) = 1/x serves as a parent rational function with a hyperbola shape. Graphing transformations such as shifts, stretches, and reflections applied to f(x) help visualize the graph of more complex rational functions like g(x).