Problem 26 - Algebra and Trigonometry Real Mathematics, Real People

Question

The zero(s) of the function are given. Verify the zero(s) both algebraically and graphically. \(Function\) \(f(x)=x-3-\frac{10}{x}\) \(Z e r o(s)\) \(x=-2,5\)

Step-by-Step Solution

Answer

Algebraically, both -2 and 5 are zeros of the function as substituting these values into the function equates to zero. Graphically, the function intersects the x-axis at points \(x=-2\) and \(x=5\), thus confirming that -2 and 5 are indeed zeros of the function.

Step 1: Verify Zero Algebraically: x = -2

Plug \(x=-2\) into \(f(x)\) and check whether the result equals zero. If the equality holds, -2 is a zero of the function. The expression is as follows: \[f(-2) = -2 - 3 - \frac{10}{-2} \] Compute this calculation.

Step 2: Verify Zero Algebraically: x = 5

Plug \(x=5\) into \(f(x)\) and check whether the result equals zero. If the equality holds, 5 is a zero of the function. The expression is as follows: \[f(5) = 5 - 3 - \frac{10}{5} \] Compute this calculation.

Step 3: Verify Zeros Graphically

Now plot \(f(x)=x-3-\frac{10}{x}\) on a graph. Look at the points where the function crosses the x-axis. They should be at \(x=-2\) and \(x=5\), thus confirming that these values are indeed the zeros of the function given in the problem.

Key Concepts

Algebraic Verification of Zeros

Identifying the zeros of a function is a fundamental aspect of analyzing algebraic expressions. The zeros are the points where the function's output is exactly zero. To algebraically verify a zero, one simply substitutes the proposed zero into the function and evaluates it. If the result is zero, the input value is indeed a true zero of the function.

For example, consider the proposed zero of -2 for the function given as \( f(x) = x - 3 - \frac{10}{x} \). By substituting -2 into the function, we get \( f(-2) = -2 - 3 + 5 \), which simplifies to 0, indicating that -2 is truly a zero of the function. On the contrary, if we substitute 5 into the function, we get \( f(5) = 5 - 3 - 2 \), which also simplifies to 0. This confirmatory process is a convincing way to ensure that the calculated zeros are correct.

The advantage of the algebraic method is that it provides concrete proof of the zeros. It's also a reliable way to check solutions derived from other methods, like factoring or using the quadratic formula.

Graphical Verification of Zeros

Graphical representations give a visual perspective of where the function intercepts the x-axis, which corresponds to the zeros of the function. This methodology is particularly useful for understanding the behavior of the function near its zeros.

The function \( f(x) = x - 3 - \frac{10}{x} \) can be graphed using a cartesian plane. The points where the graph crosses the x-axis indicate the function's zeros. For precise interpretation, it helps to plot several points of the function or use graphing technology. This is how you may confirm whether -2 and 5 are zeros as the graph must pass through these x-values.

Graphing is especially helpful for functions that are difficult to solve algebraically, or when you are looking for an approximate location of zeros. However, it's less accurate compared to algebraic verification, because the precision depends on the scale and resolution of the graph.

Rational Functions Zeros

Rational functions, which are quotients of two polynomials, often have zeros that need careful consideration. The zero is found by setting the numerator equal to zero and solving for x, assuming the denominator is not zero at the same value (which would be an undefined point or a hole rather than a zero).

In the context of the given function \( f(x) = x - 3 - \frac{10}{x} \), the zeros are determined by the expression in the numerator. However, this rational function is unique because it includes terms both outside and within a fraction. Identifying zeros requires us to equate the entire function to zero and solve for x. For \( f(x) = x - 3 - \frac{10}{x} \), this implies both the terms x-3 and -\frac{10}{x} become critical in finding where the graph touches the x-axis.

Cautions involve checking that the proposed zeros do not cause division by zero, which would indicate a vertical asymptote rather than a zero. This exercise improves comprehension by illustrating how functions with more complex structures can still have readily identifiable zeros through both algebraic and graphical exploration.