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Chapter 2: Problem 19

Determine whether the equation represents \(y\) as a function of \(x\). \(x^{2}+y=4\)

Short Answer

Expert verified
Yes, the given equation \(x^{2}+y=4\) represents \(y\) as a function of \(x\).

Step by step solution

01

Isolate y

To determine whether the equation represents \(y\) as a function of \(x\), isolate \(y\) by moving all other elements to the other side of the equation. In this case, subtract \(x^{2}\) from both sides to isolate \(y\). The new equation is \(y=4-x^2\).
02

Analyze the Result

We examine the resulting function. The equation \(y=4-x^{2}\) has one \(y\) value for every \(x\) value. Hence, \(y\) is a function of \(x\).
03

The Function's Graph

The graph of this function would also be a parabola, which opens downwards. This can be visualised as a single output for each valid input.

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Key Concepts

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

Function of x
In mathematics, when we talk about a function of x, symbolically written as f(x), we're referring to a relationship between two variables, x and y, where each input x is associated with one and only one output y. Think of it as a machine: for every item (x) you put into this machine, you get a specific item (y) out.

This concept is crucial in many areas of math because it helps us understand how one quantity changes with another. In our exercise, the equation is restructured to show y as a function of x, providing clarity on what y value corresponds to each x value.
Isolate y
Isolating y in an equation means to manipulate the equation so that y is by itself on one side of the equation. This is an essential step for identifying functions, solving equations, and graphing. The process often involves adding, subtracting, multiplying, or dividing both sides of the equation by the same number.

For instance, in our exercise, we started with the equation \(x^2 + y = 4\). By subtracting \(x^2\) from both sides, we've isolated y and got \(y = 4 - x^2\). This new form clearly shows that for every x-value, there's one y-value, which means y is a function of x.
Graph of a Function
The graph of a function is a visual representation of all the possible (x, y) pairs that satisfy the function's equation. In other words, it's a picture showing how y depends on x.

You can plot a graph on coordinate axes, where the horizontal axis (x-axis) represents the input values (x), and the vertical axis (y-axis) represents the output values (y). By plotting points and connecting them, you can see the shape of the function and any patterns it might have. For example, the graph from our exercise would show how y decreases as x increases due to the negative square of x.
Parabola
A parabola is a specific shape of a graph that resembles a U or an upside-down U. It represents the graph of quadratic functions, which have an equation in the form of \(y = ax^2 + bx + c\). The sign of the \(a\) value determines whether the parabola opens upwards or downwards.

In our scenario, since our function is \(y = 4 - x^2\), which is similar to \(y = -x^2\) when considering the parabolic motion, it means the 'a' value is negative. Due to this, the graph of our function is a parabola that opens downwards. Each value of x gives us one value of y, and thus the shape is consistent with the definition of a function.

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Most popular questions from this chapter

Evaluate the function for \(f(x)=2 x+1\) and \(g(x)=x^{2}-2\) \((f-g)(-2)\)

Describe the sequence of transformations from \(f(x)=\sqrt[3]{x}\) to \(y\). Then sketch the graph of \(y\) by hand. Verify with a graphing utility. \(y=\sqrt[3]{x+1}-1\)

Find the inverse function of the function \(f\) given by the set of ordered pairs. \(\\{(6,2),(5,3),(4,4),(3,5)\\}\)

Find (a) \(f \circ g\) and (b) \(g \circ f\). \(f(x)=2 x-3, \quad g(x)=2 x-3\)

Describe the sequence of transformations from \(f(x)=\sqrt{x}\) to \(g\). Then sketch the graph of \(g\) by hand. Verify with a graphing utility. \(g(x)=\sqrt{-x}+1\)
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