Problem 52 - Algebra 1: Concepts and Skills

Question

Find the domain of the function. $$ y=\frac{10-x}{7-x} $$

Step-by-Step Solution

Answer

The domain of the function \(y=\frac{10-x}{7-x}\) is all real numbers except 7.

Step 1: Set the Denominator equal to Zero

Firstly, to find the value that is not included in the domain, set the denominator of the fraction (7-x) equal to zero and solve for 'x'. Here it goes: 7 - x = 0

Step 2: Solve for x

When you solve above equation, you would find that x equals to 7 cause that makes the denominator zero.

Step 3: Specify the Domain

The domain of the function will be all real numbers except for the one that makes the denominator equal to zero. So, the domain of the given function is every real number except 7.

Key Concepts

Rational Functions

Rational functions are mathematical expressions that feature one polynomial divided by another. Think of them like fractions, but instead of simple numbers on top and bottom, we have expressions with variables, like \(\frac{10-x}{7-x}\). Rational functions show up everywhere in mathematics.
They help us understand relationships where variables are involved in a more complex manner than just simple addition or multiplication.A rational function is written as \(f(x) = \frac{P(x)}{Q(x)}\), where \(P(x)\) and \(Q(x)\) are polynomials. In these functions, understanding the role of the denominator is crucial:

• The numerator and the denominator can each be any polynomial function.
• The denominator cannot be zero, as division by zero is undefined in mathematics.

Rational functions can often be simplified, similar to how fractions can be reduced to their simplest form. However, unlike fractions with just numbers, it's essential to remember restrictions on values that will make the denominator zero.

Excluding Values

To find the domain of a rational function, you need to determine which values will make the denominator zero and exclude them.
In simpler terms, the domain is all the possible values of \(x\) that you can input into the function without causing an error.Here's how you do it step-by-step:

• Take the denominator of the rational function, and set it equal to zero.
• Solve the resulting equation for \(x\)
• The solutions give the values to exclude from the domain, as they make the function undefined.

For our example: \((7-x)\) is the denominator. Setting \(7-x = 0\) and solving yields \(x = 7\). Thus, the value \(x=7\) is not in the domain.

Denominator Zero

The reason why a rational function cannot have a zero in the denominator is because it leads to indeterminacy. When trying to compute something like \(\frac{a}{0}\), you quickly realize that it doesn't make sense. Mathematically, division by zero is undefined, leading to an error in calculations.
This concept is crucial in ensuring that any function we work with remains well-defined and valid for all values within its domain.Whenever you identify a value that causes the denominator to be zero, you need to rethink its place in computations.

• For the rational function \(\frac{10-x}{7-x}\), setting the denominator to zero reveals \(x=7\).
• When \(x=7\), the expression \(7-x\) becomes zero, which is problematic.

Thus, when expressing the domain, always state, "All real numbers except those that make the denominator zero," ensuring you handle any potential errors.