Problem 67 - Algebra and Trigonometry

Question

Does \(x^{2}+y=10\) define \(y\) as a function of \(x ?\)

Step-by-Step Solution

Answer

Yes, \(x^{2}+y=10\) defines \(y\) as a function of \(x\), with the function being \(y=10-x^{2}\).

Step 1: Rewrite the equation

Isolate \(y\) on one side of the equation, by subtracting \(x^{2}\) from both sides. The equation becomes \(y=10-x^{2}\).

Step 2: Check the definition of function

A function from \(x\) to \(y\) means that for each \(x\) there corresponds exactly one \(y\). But from the equation in step 1 \(y=10-x^{2}\), it's clear that for each value of \(x\) there exists exactly one value of \(y\). Thus \(y\) is a function of \(x\) defined by \(y=10-x^{2}\).

Key Concepts

Isolating the Variable

Isolating a variable in an equation is an essential skill in algebra. To 'isolate' means to get the variable by itself on one side of the equation. This helps to solve equations or to express one variable in terms of others.

In the given problem, starting with the equation \( x^2 + y = 10 \), we're tasked with expressing \( y \) in terms of \( x \). To do this, we need to isolate \( y \). By subtracting \( x^2 \) from both sides of the equation, we simplify it to \( y = 10 - x^2 \).

Here is the step clearly laid out:

• Start with the original equation: \( x^2 + y = 10 \)
• Subtract \( x^2 \) from both sides: \( y = 10 - x^2 \)

Now \( y \) is isolated, meaning we've solved for \( y \) in terms of \( x \). This change is often the first step in determining if a given formula defines a function from one variable to another.

Function Definition

Understanding what a function is can greatly simplify working with equations. A function is a relationship between two variables, typically \( x \) (the input) and \( y \) (the output), where each input corresponds to exactly one output.

After isolating \( y \) in the equation \( y = 10 - x^2 \), we can now determine if it represents a function. If for every \( x \) value, there is exactly one \( y \) value, then \( y \) is functionally dependent on \( x \). Here, when you plug in any number for \( x \), you'll get a single result for \( y \).

Here's how you can check if it fits the function criteria:

• Pick a value for \( x \).
• Substitute it into the equation \( y = 10 - x^2 \).
• Observe that you get only one \( y \) value for each \( x \) value.

In the given problem, the relation is defined by a straightforward equation with no complexities like square roots or absolute values that can produce multiple outputs for the same input.

Quadratic Equation

A quadratic equation is a polynomial equation in a single variable where the highest exponent of the variable is 2. The general form is \( ax^2 + bx + c = 0 \). In this form, the term \( ax^2 \) makes it quadratic.

In our equation \( y = 10 - x^2 \), the term \( -x^2 \) hints at quadratic behavior, even though we don't see it on the left side of the equation in the standard quadratic form. Here, the equation is simply set up to express \( y \) instead of equating it to zero.

Quadratic equations have some unique characteristics:

• They graph as parabolas.
• Depending on the sign of \( a \), they open upwards or downwards (downwards if \( a \) is negative, as with \( -x^2 \)).
• They have a vertex and may have zero, one, or two real roots.

In our example, the expression \( 10 - x^2 \) describes a downward-opening parabola. For the purposes of the function definition, the quadratic nature doesn't alter the fundamental one-to-one relationship we've determined for \( y = 10 - x^2 \). What matters is that for every \( x \), there is still a singular output \( y \), maintaining the definition of a function.