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Chapter 8: Problem 70

Use a graphing calculator to graph each function and find solutions of \(f(x)=0 .\) Then solve the inequalities \(f(x)<0\) and \(f(x)>0\). $$f(x)=x-\sqrt{x}, x \geq 0$$

Short Answer

Expert verified
Solutions: \( x = 1 \) Inequalities: \( f(x) < 0 \) for \( 0 \leq x < 1 \); \( f(x) > 0 \) for \( x > 1 \).

Step by step solution

01

- Graph the Function

Use a graphing calculator to graph the function. Enter the function \[ f(x) = x - \sqrt{x} \] and restrict the domain to \( x \geq 0 \).
02

- Find Where f(x) = 0

Identify the points where the graph intersects the x-axis. These points are the solutions to the equation \[ f(x) = 0. \] From the graph, note the x-value where this occurs.
03

- Solve f(x) = 0 Analytically

Set the function equal to zero and solve for \(x\): \[ x - \sqrt{x} = 0 \] Rewriting, we have \[ x = \sqrt{x}. \] Square both sides to remove the square root: \[ x^2 = x \] Rearrange to form a quadratic equation: \[ x^2 - x = 0 \] Factorize the equation: \[ x(x-1) = 0 \] Solve for \(x\) to get \( x = 0 \) and \( x = 1 \). Since \( x \geq 0 \), the solution is \( x = 1 \).
04

- Solve f(x) < 0

Identify the regions where the function is below the x-axis from the graph. These are the intervals where \[ f(x) < 0. \] From the graph, notice that this occurs in the interval \( 0 \leq x < 1 \).
05

- Solve f(x) > 0

Identify the regions where the function is above the x-axis from the graph. These are the intervals where \[ f(x) > 0. \] From the graph, note that this occurs in the interval \( x > 1 \).

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Key Concepts

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

Graphing Functions
Graphing functions visually represents mathematical equations. This helps you quickly identify key features like roots, intercepts, and behavior over certain intervals. When using a graphing calculator, ensure you input the function correctly. For our function, \( f(x) = x - \sqrt{x} \), input this in your calculator. Make sure to adjust the settings to only show where \( x \geq 0 \).
This will correctly display the function, highlighting where it crosses or touches the x-axis.
Quadratic Equations
A quadratic equation is a second-degree polynomial typically in the form \( ax^2 + bx + c = 0 \). In this exercise, we derived a quadratic equation by setting \( f(x) = 0 \). We had: \[ x - \sqrt{x} = 0 \]
Squaring both sides gives \( x^2 = x \), which rearranges to \( x^2 - x = 0 \)
This can be solved by factoring, which we'll discuss next.
Factoring
Factoring breaks down complex equations into simpler components. In our quadratic equation \( x^2 - x = 0 \), we find the common factors: \( x(x-1) = 0\). The solutions are obtained when each factor equals zero. Thus:\( x = 0 \) \( x = 1 \).
Since our domain constraint is \( x \geq 0 \), only \( x = 1 \) serves as a valid solution. This tells us where the function \( f(x) \) meets the x-axis.
Inequalities
Inequalities describe where one value lies in relation to another. To solve \( f(x) < 0 \) and \( f(x) > 0 \), observe the graph's behavior relative to the x-axis. From the graph, \( f(x) < 0 \) where \( 0 \leq x < 1 \). The function drops below the x-axis in this range. For \( f(x) > 0 \), find where the function rises above the x-axis: \( x > 1 \). These intervals pinpoint when the function holds these inequalities.
Constraints on Domain
Constraints on domain limit the values \( x \) can take. For \( f(x) = x - \sqrt{x} \), the constraint is \( x \geq 0 \). This is crucial because \sqrt{x} isn't defined for negative values. Therefore, always graph within the defined domain. If ignored, the graph and solutions may be incorrect or misleading. Reviewing and applying constraints accurately ensures valid solutions and graph interpretations.

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Most popular questions from this chapter

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Use a graphing calculator to graph each function and find solutions of \(f(x)=0 .\) Then solve the inequalities \(f(x)<0\) and \(f(x)>0\). $$f(x)=\frac{x^{3}-x^{2}-2 x}{x^{2}+x-6}$$
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