Use the given functions f and g. (a) Solve f(x)=0 (b) Solve...

Use the given functions f and g.
(a) Solve $f(x)=0$
(b) Solve $g(x)=0$
(c) Solve $f(x)=g(x)$
(d) Solve $f(x)>0$
(e) Solve $g(x) \leq 0$
(f) Solve $f(x)>g(x)$
(g) Solve $f(x) \geq 1$
$$\begin{array}{l}f(x)=x^{2}-x-2 \\g(x)=x^{2}+x-2\end{array}$$

Key Concepts

Quadratic Equations

Quadratic equations involve polynomials of degree 2 and are typically solved by techniques such as factoring, completing the square, or applying the quadratic formula. They form the basis for finding the roots or zeros of quadratic functions, which represent the values at which the function equals zero.

Factoring Techniques

Factoring is a method used to rewrite quadratic expressions as a product of simpler linear factors. This approach makes it straightforward to resolve quadratic equations by setting each factor equal to zero, which is crucial when solving equations like f(x)=0 or g(x)=0.

Quadratic Inequalities and Sign Analysis

Quadratic inequalities require determining the intervals where a quadratic expression is greater than or less than a certain number. Sign analysis, often done by testing values in different intervals defined by the roots, is a key technique to ascertain where the function takes positive or negative values.

Function Intersection and Comparison

Finding the intersection of functions involves equating two functions to identify common points, while comparing functions typically involves analyzing the difference between them. This concept is important when solving equations like f(x)=g(x) or inequalities such as f(x) > g(x), as it reduces the problem to studying a single function or its difference.

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