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Chapter 1: Problem 69

In Exercises 67-74, graph the function and determine the interval(s) for which \(f(x) \geq 0\). \(f(x) = 9-x^2\)

Short Answer

Expert verified
The function \(f(x) = 9 - x^2\) is greater than or equal to zero on the interval \([-3, 3]\).

Step by step solution

01

Graphing the function

Plot the function \(f(x) = 9 - x^2\) on a graph. You might first want to calculate a few data points to get an initial idea of the graph. You'll find out that there are two points on the x-axis where \(f(x) = 0\), these points are \(+3\) and \(-3\). This is a parabola that opens downwards.
02

Determine where the function is non-negative

From the graph, observe the parts of the graph that are on or above the x-axis since these are the points where \(f(x) \geq 0\). You'll observe that the parts of the function on or above the x-axis are between \(x = -3\) and \(x = 3\). In other words, the parabola is above x-axis when \(x\) is in the interval from \(-3\) to \(3\).
03

Write down the interval

Finally, write down the interval in which the function is non-negative. This is on the closed interval \([-3, 3]\), where both endpoints are included (as the function is zero there, and hence also greater or equal to zero).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graphing Parabolas
When graphing parabola functions, it鈥檚 important to understand their shape and direction. A parabola is a U-shaped curve that can either open upwards or downwards. For the function \(f(x) = 9 - x^2\), the parabola opens downwards because the coefficient of \(x^2\) is negative.

To sketch this parabola:
  • Identify the vertex, which is the highest point for a downward-opening parabola. Here, the vertex is at \((0, 9)\), because \(9 - 0^2 = 9\).
  • Calculate where the parabola crosses the x-axis by setting \(f(x) = 0\). Solve for \(x\) in \(9 - x^2 = 0\) to get \(x = 3\) and \(x = -3\).
  • Plot these points and the vertex on a graph to visualize the curve.
By following these steps, you can clearly see the shape and position of a parabola. This provides a visual foundation for understanding other concepts like inequalities and intervals.
Inequalities
Inequalities help us understand when a function holds certain values compared to a given number. In this exercise, we're looking for the values of \(x\) where the function is greater than or equal to zero, or \(f(x) \geq 0\).

For our function \(f(x) = 9 - x^2\):
  • When \(x\) is between \(-3\) and \(3\), \(f(x)\) is on or above the x-axis.
  • This means \(f(x)\) is positive or zero within this range.
  • For the values outside of this range, \(f(x)\) becomes negative, meaning it dips below the x-axis.
By solving \(9 - x^2 \geq 0\), you can understand the nature of the graph and the intervals where the function maintains non-negative values.
Intervals
Understanding intervals is crucial when working with functions and inequalities. Intervals describe a range of values within which certain conditions hold true.

For the function \(f(x) = 9 - x^2\), we identified that \(f(x)\) is non-negative between \(x = -3\) and \(x = 3\). This gives us a closed interval, written as \([-3, 3]\).

Why is it closed?
  • A closed interval includes its endpoints. At \(x = -3\) and \(x = 3\), \(f(x) = 0\), meaning the function equals zero and satisfies \(f(x) \geq 0\).
  • This tells us that every value from \(-3\) to \(3\), inclusive, is part of the solution.
Knowing how to read and write intervals allows you to express conditions on the graph in a precise and mathematical way, simplifying the communication of the solution.

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