91Ó°ÊÓ

To grasp the concept of zeros in rational functions, we must first understand what zeros represent in general. The zero of a function, sometimes referred to as the root, is the value of the variable that results in the function equating to zero. In simple terms, if you substitute a particular value of x into a function and the outcome is zero, then that value is a zero of the function.

For rational functions, like the one given in the exercise, zeros are calculated in the same way as other types of functions. If you consider the function:

• \(f(x) = 1 - \frac{2}{x-7}\)

To find the zero of this function, you set it equal to zero and solve for x. It's crucial to note that zeros only make sense in the domain of the function, so you cannot include values that would make any denominator zero. In this example, the function would be undefined at \(x = 7\), so this cannot be a zero.

Solving equations is a fundamental skill in mathematics and is crucial for finding zeros of functions. In the exercise, we are given a rational function \(f(x) = 1 - \frac{2}{x-7}\) and tasked with finding the values of x that make \(f(x) = 0\).

The process begins with setting the equation equal to zero. From

• \(0 = 1 - \frac{2}{x-7}\)

The next step involves isolating x by performing algebraic manipulations. Start by rearranging the equation:

• Add \(\frac{2}{x-7}\) to both sides resulting in \(1 = \frac{2}{x-7}\).
• Multiply both sides by \(x - 7\) to eliminate the fraction, leading to \(x - 7 = 2\).
• Finally, solve for x by adding 7 to both sides to get \(x = 9\).

This solution shows the value of x where the function equals zero, meaning x = 9 is the solution to the equation.

Verification is a vital step when solving any mathematical problem as it ensures the solution is indeed correct. After finding solutions, such as zeros of a function, it is important to substitute them back into the original function to confirm they satisfy the equation.

In our exercise, we found the zero of the function \(f(x) = 1 - \frac{2}{x-7}\) is x = 9.
Now, to verify:

• Substitute x = 9 back into the function.
• Calculate \(f(9) = 1 - \frac{2}{9-7}\).
• Simplifying gives \(f(9) = 1 - 1 = 0\).

Since we obtain zero, our value of x = 9 is indeed a correct zero of the function. This verification step not only checks for algebraic errors but also reconfirms that the solution lies within the domain of the function.