Problem 25 - Algebra and Trigonometry Real Mathematics, Real People

Question

The zero(s) of the function are given. Verify the zero(s) both algebraically and graphically. \(Function\) \(f(x)=\frac{x+1}{2}-\frac{x-2}{7}+1\) \(Z e r o(s)\) \(x=-5\)

Step-by-Step Solution

Answer

After algebraically substituting \(x\) with -5 in the equation, the result is found to be zero, thus confirming that -5 is a zero of the function. A graphical representation of the function also confirms this as the curve touches the x-axis at \(x=-5\). Therefore, both algebraically and graphically, \(x=-5\) is verified as a zero of the function

Step 1: Substitute \(x\) with -5

Substitute \(x\) with -5 in the function \(f(x)=(x+1)/2 - (x-2)/7 + 1\). If the result equals zero, then -5 is indeed a zero of the function.

Step 2: Compute the Result

Perform the required operations to compute the result. After substituting \(x\) with -5, the function becomes \((-5+1)/2 - ((-5)-2)/7 + 1\). After calculating, the result is found to be zero, thus confirming algebraically that \(x=-5\) is a zero of the function.

Step 3: Graphical Verification

The function needs to be graphed and it should be checked whether the line crosses the x-axis at \(x=-5\). Visualizing the curve of the function can provide an additional verification for the zero. Using any graphing tool would result in a graph of the function and this graph should hit the x-axis at \(x=-5\), visually confirming that -5 is indeed a zero of the function.

Key Concepts

Algebraic Verification of Zeros

Algebraic verification is a mathematical procedure used to determine whether a given value is indeed a zero of a function. This means checking if substituting this value into the function results in an output of zero. To perform algebraic verification for a function like
\[\begin{equation}f(x) = \frac{x+1}{2} - \frac{x-2}{7} + 1\end{equation}\]
, follow these steps: First, replace the variable 'x' with the value in question—in this case, -5. The resulting expression should then be simplified. If the simplified expression equals zero, the verification is complete, and you can conclude that -5 is a zero of the function. When performing the calculation, it's crucial to apply the arithmetic operations correctly, including the order of operations, to avoid errors that could lead to incorrect conclusions.

Graphical Verification of Zeros

Graphical verification is quite intuitive; it involves visual inspection of the graph of a function to confirm its zeros. For the function given in the problem, you would plot the graph using a graphing tool or by sketching it manually based on calculated points. The function is described as
\[\begin{equation}f(x) = \frac{x+1}{2} - \frac{x-2}{7} + 1\end{equation}\]
To graphically verify that -5 is a zero, find the point on the graph where the function crosses the x-axis. This intersection represents a zero because the y-value (output) at that point is zero. If the graph passes through the point (-5, 0), then the graphical verification is successful. It's important to note that any discrepancies between algebraic and graphical verifications should prompt a review of both methods to identify possible errors.

Understanding Rational Functions

A rational function is a fraction where both the numerator and the denominator are polynomials. The function
\[\begin{equation}f(x) = \frac{x+1}{2} - \frac{x-2}{7} + 1\end{equation}\]
is an example of a rational function once simplified. These functions are interesting because they can have discontinuities (such as holes or vertical asymptotes) where the denominator equals zero. However, in the context of finding zeros, we are only concerned with the values of 'x' that make the entire function equal to zero—that is, where the function's graph intersects the x-axis. When working with rational functions, remember to look out for common misconceptions like confusing zeros with asymptotes. Zeros are the x-values that result in the output being zero, whereas asymptotes are lines that the graph approaches but never actually reaches.