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Chapter 2: Problem 18

Find a single function defined implicitly by the given equation. Graph the function and give its domain. Use a graphing utility if necessary.] $$ -x+\sqrt{y}+2=0 $$

Short Answer

Expert verified
The function defined is \(y = (x - 2)^2\), with domain \(x \in \mathbb{R}\).

Step by step solution

01

Solve for y in Terms of x

Rearrange the given implicit equation \(-x + \sqrt{y} + 2 = 0\) to solve for \(y\). Start by isolating \(\sqrt{y}\) on one side: \[\sqrt{y} = x - 2\]. Now, square both sides to solve for \(y\): \[y = (x - 2)^2\]. This is the explicit function in terms of \(x\).
02

Define the Domain

The expression \((x - 2)^2\) is defined for all real numbers because any real number squared gives a non-negative result, which means \(y\) can take any non-negative value. However, since the square root \(\sqrt{y}\) in the implicit equation must have been non-negative, \(y\) must also be non-negative. Thus, the explicit function \(y = (x-2)^2\) is valid for all real \(x\). The domain of \(y = (x - 2)^2\) is all real numbers, \(x \in \mathbb{R}\).
03

Graph the Function

Graph the function \(y = (x - 2)^2\). This is a parabola opening upwards with its vertex located at the point \((2, 0)\). This graph illustrates all points \((x, y)\) satisfying the equation, showing that any real \(x\) results in a valid non-negative \(y\) value.
04

Use a Graphing Utility if Necessary

If a graphing calculator or computer software is available, use it to create a precise graph of the function \(y = (x-2)^2\). This can help in visualizing the parabola and verifying the domain and range conclusions.

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Key Concepts

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

Function Graphing
Graphing functions helps us visualize how the values change. To graph a function, you plot points in an x-y plane based on the function鈥檚 equation. For our equation, we first solve for one variable in terms of others to get an explicit function like \( y = (x-2)^2 \).
  • This function is called an explicit function because \( y \) is given directly as a formula in terms of \( x \).
  • Creating a graph involves plotting points derived from values you plug into the function.
Graphing \( y = (x-2)^2 \) involves creating a curve by connecting points like \( (2,0) \), \( (1,1) \), and \( (3,1) \), deriving a parabola. This visual representation helps us explore features like symmetry, intercepts, and the shape of the curve.
Domain of a Function
Understanding the domain is crucial as it tells us all possible x-values a function can accept. For the function \( y = (x-2)^2 \), the domain is all real numbers.
  • This is because there's no restriction on \( x \) in the equation; any real number plugged in results in a valid \( y \) value.
  • Since the function involves squaring \((x-2)\), it naturally handles both positive and negative \( x \) values.
In general, whenever you have polynomials like this, they often cover the entire set of real numbers unless specified by other constraints like division by zero or square roots of negative numbers.
Solving Equations
Solving equations involves finding the values of variables that satisfy given conditions. In implicit equations, both variables are initially on the same side. For \( -x + \sqrt{y} + 2 = 0 \), we rearrange it to find an explicit expression for \( y \).
  • First, isolate \( \sqrt{y} \) by adding \( x \) and subtracting 2 from both sides, yielding \( \sqrt{y} = x - 2 \).
  • Next, square both sides to solve for \( y \): \( y = (x - 2)^2 \).
This step is key for transforming implicit forms into explicit ones, making it easier to identify the function鈥檚 behavior and characteristics.
Parabolic Functions
Parabolic functions, like \( y = (x-2)^2 \), form a specific category of functions with distinct features.
  • They graph as U-shaped curves,
    • which can open upwards or downwards based on the sign of the squared term.
  • Our function's parabola opens upwards, indicated by the positive coefficient of the \( (x-2)^2 \) term.
  • The vertex of \( y = (x-2)^2 \) is at \((2, 0)\), showing the lowest point in this case.
Understanding parabolic functions is essential due to their repetitive occurrence in math and real-world applications, such as projectile motion and quadratic equations.

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Most popular questions from this chapter

In Problems \(9-34\), sketch the graph of the given piece wise-defined function. Find any \(x\) - and \(y\) intercepts of the graph. Give any numbers at which the function is discontinuous. $$ y=\left\\{\begin{array}{ll} 1, & x<0 \\ |x-1|, & 0 \leq x \leq 2 \\ 1, & x>2 \end{array}\right. $$

In Problems \(1-4\), find the indicated values of the given piecewise-defined function \(f\). $$ f(x)=\left\\{\begin{array}{ll} x^{2}+2 x, & x \geq 1 \\ -x^{3}, & x<1 ; \end{array} \quad f(1), f(0), f(-2), f(\sqrt{2})\right. $$

The given function \(f\) is one-to one. Find \(f^{-1}\). Sketch the graphs of \(f\) and \(f^{-1}\) on the same rectangular coordinate system. $$ f(x)=2-\sqrt{x} $$

In Problems \(9-34\), sketch the graph of the given piece wise-defined function. Find any \(x\) - and \(y\) intercepts of the graph. Give any numbers at which the function is discontinuous. $$ y=\left|x^{2}-2 x\right| $$

In Problems \(9-34\), sketch the graph of the given piecewise-defined function. Find any \(x\) - and \(y\) intercepts of the graph. Give any numbers at which the function is discontinuous. $$ y=\left\\{\begin{array}{ll} -x, & x \leq 1 \\ -1, & x>1 \end{array}\right. $$
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