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 $f(x) \geq 1$
$$\begin{aligned}&f(x)=x^{2}-4\\&g(x)=-x^{2}+4\end{aligned}$$

Key Concepts

Intersection of Functions

Finding the intersection of functions means setting two function expressions equal to each other and solving for the variable. This concept is important when comparing functions, determining where one function equals or exceeds another, and understanding the relationship between different algebraic expressions.

Solving Quadratic Inequalities

Solving quadratic inequalities requires determining the intervals where the quadratic expression is positive, negative, or meets specific inequality conditions. This usually involves identifying the roots of the related quadratic equation and analyzing the sign of the expression on intervals defined by those roots.

Quadratic Functions

Quadratic functions are second-degree polynomial functions typically written in the form ax² + bx + c. They are characterized by parabolic graphs and key features such as the vertex, axis of symmetry, and roots. Understanding these functions is fundamental to analyzing their behavior and solving related equations and inequalities.

Solving Quadratic Equations

Solving quadratic equations involves finding the values of x for which the quadratic expression equals zero. Common methods include factoring, using the quadratic formula, and completing the square. This process is essential to determine the zeros of the quadratic function, which are often used as boundary points in further analysis.

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