Question

If \(0x^2 $$ \(x

Short answer

For the interval \(0 x^2\) is true.

Step-by-step solution

Graph Function Analysis

Draw the graphs of $f(x)=x$ and $g(x)=x^2$. The shape of the curves helps us to visualize the problem and understand the behavior of these functions between 0 and 1.

Point Comparison

Compare the points on the functions $f(x)=x$ and $g(x)=x^2$. For \(0 x^2\) for $0<x<1$.

Conclusion

Based on the visual inspection of the graphs and the value comparison, it can be concluded that $x>x^2$ for the given interval $0<x<1$.

Key concepts

Graphical Analysis

Graphical analysis is a powerful way to understand the relationship between different mathematical functions. By sketching the graphs of functions, you can easily identify patterns, intersections, and differences. For the problem at hand, we drew the graphs of two functions:

• Function 1: $f(x)=x$
• Function 2: $g(x)=x^2$

Both graphs are parabolas, but they behave differently between 0 and 1. When graphing these functions, $f(x)=x$ appears as a straight line passing through the origin (0,0) and points toward (1,1), while $g(x)=x^2$ is a curve that bends downwards more steeply.
By looking at the graphs, you can see that between 0 and 1, the line representing $f(x)=x$ remains above the curve representing $g(x)=x^2$, demonstrating that $x>x^2$ within this range. Graphical analysis simplifies comparing these functions, offering a visual confirmation of their behavior.

Quadratic Functions

Quadratic functions are polynomial functions of degree 2, typically represented as $f(x)=a{x}^2+bx+c$. In our given problem, we focused specifically on the simplest quadratic form $g(x)=x^2$.
Quadratic functions such as $g(x)=x^2$ have distinctive U-shaped curves called parabolas. The coefficient $a$ is positive here, meaning the parabola opens upwards.
It's essential to understand the behavior of the parabola between 0 and 1, as this is where our interest lies.

• When $x=0.5$, $g(0.5)=(0.5{)}^2=0.25$, illustrating that $x^2$ is smaller than $x$.
• The purpose of this analysis shows that for any $x$ between 0 and 1, squaring $x$ yields a value closer to zero, emphasizing that $x>x^2$ when $0<x<1$.

Quadratic functions are foundational in mathematics, appearing in various fields such as physics, engineering, and economics, making understanding them deeply valuable.

Interval Notation

Interval notation is a concise way of expressing a range of numbers in mathematics. It's particularly useful when defining the domain of functions or the solutions to inequalities. In our example, the interval $0<x<1$ is utilized, which includes all numbers between 0 and 1 but not the endpoints themselves.
When using interval notation:

• Parentheses $()$ indicate that an endpoint is not included (open interval).
• Brackets $[]$ indicate that an endpoint is included (closed interval).

So, $(0,1)$ means that $x$ is greater than zero but less than one. This notation is both efficient and clear, making it easier to communicate which numbers are considered in a given problem.
For the inequality problem, expressing the domain in interval notation helps clarify the specific values for which $x>x^2$ is true. Understanding and using interval notation correctly is crucial for solving many mathematical problems efficiently.