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Chapter 1: Problem 22

Determine whether the equation represents \(y\) as a function of \(x\). $$ y-4 x^{2}=36 $$

Short Answer

Expert verified
Yes, the given equation does represent \(y\) as a function of \(x\).

Step by step solution

01

Rearrange the equation to isolate y

First, the given equation \(y - 4x^{2} = 36\) has to be rearranged in terms of y. Add \(4x^{2}\) to both sides of the equation to get \(y = 4x^{2} + 36\)
02

Check your solution

Look at the given equation. It is clear that for each x-value we obtain just one value of y. Therefore, this equation does represent \(y\) as a function of \(x\)

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Key Concepts

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

Determining Functions
When working with equations in precalculus, it's vital to determine if an equation represents a function. A function maps each element in the domain (in this case, the set of all possible values of x) to exactly one element in the codomain (the potential values of y).

In the given equation, y - 4x2 = 36, our task is to decide if y is a function of x. After rearranging the equation and solving for y, we get y = 4x2 + 36. In this format, it's clear that for any value of x, there's precisely one corresponding y value, satisfying the definition of a function. This one-to-one relationship is the crux of determining functions in precalculus.
Functional Notation
Functional notation is a concise way to signify the relationship between variables in functions. It's typically written as f(x), where f denotes the function and x the variable. For example, if y = 4x2 + 36, we can express y as f(x) which becomes f(x) = 4x2 + 36.

This notation isn't just for show; it's extremely helpful for evaluating the function at specific x values, indicating composition of functions, and can be used in more complex scenarios such as derivatives and integration in calculus.
Quadratic Functions
Quadratic functions are polynomials of the second degree, which means the highest exponent of the variable is two. The general form of a quadratic function is f(x) = ax2 + bx + c, where a, b, and c are constants, and a is non-zero.

The function f(x) = 4x2 + 36, derived from the exercise, is a quadratic function because it's in the form of ax2 + c, where a = 4 and c = 36. Quadratics are characterized by their parabolic graphs, which can open upwards or downwards depending on the sign of a. In this case, the parabola opens upwards and has its vertex at the minimum point.
Equation Manipulation
Equation manipulation involves rearranging and simplifying equations, a foundational skill in precalculus. Through manipulation, we isolate variables, simplify expressions, and solve equations. In our example, to isolate y, we add 4x2 to both sides to reverse the subtraction in the original equation.

This step is essential as it converts the equation into a form that clearly shows the function's dependence on x and allows us to evaluate the function for specific x values. Mastering equation manipulation often involves practice with distribution, factoring, and dealing with fractions, radicals, and exponents.

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Most popular questions from this chapter

The mathematical model \(y=\frac{\kappa}{x}\) is an example of _____ variation.

Use the given value of \(k\) to complete the table for the direct variation model $$y=k x^{2}$$ Plot the points on a rectangular coordinate system. $$\begin{array}{|l|l|l|l|l|l|} \hline x & 2 & 4 & 6 & 8 & 10 \\ \hline y=k x^{2} & & & & & \\ \hline \end{array}$$ $$ k=\frac{1}{4} $$

Determine whether the function has an inverse function. If it does, find the inverse function. $$ f(x)=\sqrt{2 x+3} $$

In direct variation models of the form \(y=k x, k\) is called the _____ of _____ .

Determine whether the function has an inverse function. If it does, find the inverse function. $$ g(x)=\frac{x}{8} $$
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