Calculate the Area of f(x) = x²: Positive Region Analysis

Q: Why isn't the answer just x > 0?

Because negative values of x also make the function positive! When you square a negative number like (-2), you get $ (-2)^2 = 4 $, which is positive. The function $ f(x) = x^2 $ is positive for both positive and negative x values.

Q: What's special about x = 0?

At $ x = 0 $, the function equals zero: $ f(0) = 0^2 = 0 $. Since zero is neither positive nor negative, we exclude it when looking for where the function is positive.

Q: How do I remember that squaring makes numbers positive?

Think of it this way: multiplying two numbers with the same sign always gives a positive result. Since $ x^2 = x \times x $, you're always multiplying a number by itself, so the result is positive (except when x = 0).

Q: What does the graph of this function look like?

The graph of $ f(x) = x^2 $ is a U-shaped parabola that opens upward. It touches the x-axis at the origin (0,0) and rises on both sides, showing that the function is positive everywhere except at x = 0.

Q: Is there a difference between 'positive area' and 'positive values'?

In this context, 'positive area' refers to the regions where the function has positive y-values. We're looking for x-values where $ f(x) > 0 $, not calculating actual area under a curve.

Quadratic Functions with Positive Value Analysis

Find the positive area of the function

$f(x)=x^2$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find the positive domain of the function

00:03 Note the coefficient of X squared is positive, therefore the function is smiling (opens upward)

00:11 The positive domain is above the X-axis

00:14 Therefore, we substitute Y=0 to find the intersection points with the X-axis

00:20 This is the intersection point of the function with the X-axis

00:24 Let's mark the intersection points with the X-axis

00:36 The positive domain is above the X-axis

00:40 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Find the positive area of the function

$f(x)=x^2$

Step-by-step solution

To determine where the function $f(x) = x^2$ is positive, we consider the nature of this parabolic graph, which opens upwards.

Step 1: Recognize that the function $f(x) = x^2$ outputs non-negative values for any real number $x$ . The graph of this function is a U-shaped parabola.

Step 2: Analyze the values of the function:
- For $x = 0$ , $f(0) = 0^2 = 0$ .
- For $x \neq 0$ , $f(x) = x^2 > 0$ , because squaring any non-zero real number results in a positive value.

Therefore, the function is positive for all $x$ except at $x = 0$ , where it is zero.

Step 3: Based on the comparison given in the choices, and our calculation, the area of interest is positive for $x \neq 0$ .

Thus, the solution to the problem is that the positive area occurs for $x \neq 0$ .

Final Answer

$x≠0$

Key Points to Remember

Essential concepts to master this topic

Rule: Squared values are always non-negative for real numbers
Technique: Test $x^2$ at x = 0 gives 0, x ≠ 0 gives positive
Check: Verify $(-3)^2 = 9 > 0$ and $3^2 = 9 > 0$ ✓

Common Mistakes

Avoid these frequent errors

Thinking negative x values make the function negative
Don't assume $f(x) = x^2$ is negative when x < 0 = wrong conclusion! Squaring any negative number produces a positive result. Always remember that $(-a)^2 = a^2$ for any real number a.

Practice Quiz

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Which chart represents the function $$ y=x^2-9 $$ ?

FAQ

Everything you need to know about this question

Why isn't the answer just x > 0?

Because negative values of x also make the function positive! When you square a negative number like (-2), you get $(-2)^2 = 4$ , which is positive. The function $f(x) = x^2$ is positive for both positive and negative x values.

What's special about x = 0?

At $x = 0$ , the function equals zero: $f(0) = 0^2 = 0$ . Since zero is neither positive nor negative, we exclude it when looking for where the function is positive.

How do I remember that squaring makes numbers positive?

Think of it this way: multiplying two numbers with the same sign always gives a positive result. Since $x^2 = x \times x$ , you're always multiplying a number by itself, so the result is positive (except when x = 0).

What does the graph of this function look like?

The graph of $f(x) = x^2$ is a U-shaped parabola that opens upward. It touches the x-axis at the origin (0,0) and rises on both sides, showing that the function is positive everywhere except at x = 0.

Is there a difference between 'positive area' and 'positive values'?

In this context, 'positive area' refers to the regions where the function has positive y-values. We're looking for x-values where $f(x) > 0$ , not calculating actual area under a curve.

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