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

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 $f(x) \geq 1$.
$$
\begin{array}{c}
f(x)=-x^{2}+3 \\
g(x)=-3 x+3
\end{array}
$$

Key Concepts

Quadratic Functions

Quadratic functions are polynomial functions of degree two, typically written in the form ax² + bx + c. They generate parabolic graphs whose curvature and direction depend on the coefficient a. When solving equations involving quadratic functions, methods such as factoring, completing the square, or applying the quadratic formula are often used, with the discriminant indicating the nature of the roots.

Linear Functions

Linear functions are first-degree polynomial functions expressed in the form mx + b. Their graphs are straight lines, and they serve as a straightforward example of a function where the relationship between variables is constant. Solving equations involving linear functions relies on simple algebraic manipulation, making them useful as benchmarks when comparing with more complex functions like quadratics.

Solving Equations

Solving equations means finding the values of variables that make an equation true. This can involve setting a function equal to zero, determining where two functions are equal, or solving for when one function equals a specific value. The techniques used depend on the type of function involved, with linear equations typically resolving through basic algebra and quadratic equations requiring more nuanced approaches like factoring or the quadratic formula.

Solving Inequalities

Solving inequalities involves determining the set of values that satisfy a given inequality condition, such as a function being greater than or less than another function or a specific number. This process often includes identifying critical values—points where the expression equals zero—and performing sign analysis on the resulting intervals. Special care is taken with strict versus non-strict inequalities to accurately include or exclude boundary points.

Comparing Functions

Comparing functions entails finding where one function is greater than, less than, or equal to another. This is typically handled by setting up the appropriate inequality (for example, f(x) > g(x) or f(x) = g(x)) and solving it using algebraic methods. The process may involve subtracting one function from the other to reduce the problem to analyzing the sign of a single expression, allowing for a clearer understanding of the relationship between the functions across different intervals.

Transcript

Please give Ace some feedback