Let $\text{ƒ}\left(x\right)=3x^2-4$ and $g\left(x\right)=x^2-3x-4$. Find each of the following.
$\left(f/g\right)(-1)$

Verified step by step guidance

1. Identify the functions given: \(f(x) = 3x^2 - 4\) and \(g(x) = x^2 - 3x - 4\).
2. Understand that \((f/g)(x)\) means the function \(f(x)\) divided by \(g(x)\), so \((f/g)(x) = \frac{f(x)}{g(x)}\).
3. Substitute \(x = -1\) into both functions separately: calculate \(f(-1)\) and \(g(-1)\) by replacing \(x\) with \(-1\) in each expression.
4. Write the expression for \((f/g)(-1)\) as \(\frac{f(-1)}{g(-1)}\) using the values found in the previous step.
5. Simplify the fraction \(\frac{f(-1)}{g(-1)}\) by performing the arithmetic operations in the numerator and denominator, then divide to find the value of \((f/g)(-1)\).

Key Concepts

Function Notation and Evaluation

Function notation, such as f(x), represents a rule that assigns each input x to an output. Evaluating a function at a specific value means substituting that value into the function's expression and simplifying to find the output.

Function Division (Quotient of Functions)

The division of two functions (f/g)(x) is defined as f(x) divided by g(x), provided g(x) ≠ 0. To find (f/g)(a), evaluate both f(a) and g(a) separately, then divide the results.

Polynomial Functions and Simplification

Both f(x) and g(x) are polynomial functions, which involve powers of x with coefficients. Simplifying polynomial expressions after substitution involves applying arithmetic operations and combining like terms carefully.