In Exercises 31–50, find ƒ/g and determine the domain for each function. f(x) = 2x² − x − 3, g (x) = x + 1

Verified step by step guidance

1. Identify the functions: $f(x)=2x^2-x-3$ and $g(x)=x+1$.
2. The expression $\frac{f}{g}$ represents the division of the two functions, which is $\frac{f(x)}{g(x)}=\frac{2x^2-x-3}{x+1}$.
3. To determine the domain of $\frac{f}{g}$, identify the values of $x$ for which $g(x)\ne 0$.
4. Set $g(x)=x+1$ equal to zero and solve for $x$: $x+1=0\Rightarrow x=-1$.
5. The domain of $\frac{f}{g}$ is all real numbers except $x=-1$, since division by zero is undefined.

Key Concepts

Function Division

Function division involves creating a new function by dividing one function by another. In this case, we are finding ƒ/g, which means we will divide the function f(x) = 2x² − x − 3 by g(x) = x + 1. This process requires understanding how to simplify the resulting expression and identify any restrictions on the variable.

Domain of a Function

The domain of a function is the set of all possible input values (x-values) for which the function is defined. When dividing functions, it is crucial to identify values that make the denominator zero, as these values are excluded from the domain. For g(x) = x + 1, the domain will exclude x = -1, where the function is undefined.

Polynomial Functions

Polynomial functions are expressions that involve variables raised to whole number powers, combined using addition, subtraction, and multiplication. The function f(x) = 2x² − x − 3 is a quadratic polynomial, which can be analyzed for its roots and behavior. Understanding polynomial functions is essential for performing operations like division and determining their domains.