Question

What are the \(x\)-intercept(s) of the graph of the function \(y=\frac{x-5}{x^2-1}\) ? (A) \(1,-1\) (B) 5 (C) 1 (D) \(-5\)

Short answer

(B) 5

Step-by-step solution

Set the function equal to zero

To find the x-intercepts, the function \(y\) must be set equal to zero and the resulting equation solved for \(x\). This gives us the equation: \(0=\frac{x-5}{x^2-1}\)

Cross-multiply

Since this equation is in fraction equals form, cross-multiplication may be used to simplify it. By considering the denominator on the right-hand side as '1', the equation can be rewritten as: \(0*(x^2-1) = (x-5)*1\) which simplifies to:\(0 = x - 5\).

Solve the equation for \(x\)

To isolate \(x\), the constant 5 should be added to both sides of the equation to make it:\(x = 5\).

Key concepts

Rational Functions

A rational function is a type of mathematical expression defined by the ratio of two polynomials. It takes the form \( f(x) = \frac{P(x)}{Q(x)} \), where both \( P(x) \) and \( Q(x) \) are polynomials and \( Q(x) eq 0 \).

The rational function discussed here is \( y=\frac{x-5}{x^2-1} \). Here's what makes rational functions unique and important:

• **Domain Restrictions**: Because the denominator cannot be zero, we need to find which values of \( x \) would make \( Q(x) = 0 \) and exclude them from the domain.
• **Behavior at Asymptotes**: Rational functions often have vertical asymptotes (vertical lines which the graph approaches but never touches) at these excluded values. Horizontal or slant asymptotes can also occur, providing lines the graph approaches as \( x \) becomes very large or very small.
• **Intercepts and Turning Points**: These are critical points where the function may cross the axes or change direction. The \( x \- ext{intercepts} \) are found where \( y = 0 \).

Understanding rational functions is vital for recognizing patterns in many real-world contexts, from physics to economics.

Cross-Multiplication

Cross-multiplication is a technique mainly used to solve equations involving fractions. It simplifies the process by eliminating the fractions, making the equations easier to solve. Here’s how cross-multiplication works:

Consider an equation where a fraction is equal to another number or fraction: \( \frac{a}{b} = \frac{c}{d} \). By cross-multiplying, you multiply both sides of the equation by the denominators, resulting in \( a \cdot d = b \cdot c \).

• **Purpose**: This method allows us to get rid of denominators, reducing the equation to a simpler, more straightforward form.
• **Application**: In our exercise, we started with \( 0=\frac{x-5}{x^2-1} \). By treating the denominator \( x^2-1 \) as equal to 1 on the other side, the equation was simplified to \( 0 \cdot (x^2-1) = (x-5) \), which is \( 0 = x-5 \).

Cross-multiplication is particularly useful for dealing with rational functions where interpreting results and reducing complexity is key.

Solving Equations

Solving an equation involves finding the value(s) of the variable that make the equation true. This process can vary in difficulty from simple algebraic manipulations to more complex, multi-step solutions.

For the given problem, after cross-multiplying, we arrived at the equation \( 0 = x - 5 \). The next step is solving this simple equation. The process generally involves:

• **Isolating the Variable**: Aim to have the variable (in this case, \( x \)) on one side of the equation.
• **Simplification**: Perform arithmetic operations to both sides to keep the equation balanced. Here, we added 5 to both sides, yielding \( x = 5 \).

The solution to the equation \( x = 5 \) indicates that the x-intercept of the function \( y=\frac{x-5}{x^2-1} \) is at \( x = 5 \), which corresponds to option (B) in the original problem.

Algebraic Manipulation

Algebraic manipulation involves using various algebraic properties and operations to simplify, rearrange, or solve expressions or equations. It is a vital skill in handling rational functions and many other types of mathematical problems.

Here, algebraic manipulation was required to solve \( 0 = \frac{x-5}{x^2-1} \). Important techniques include:

• **Rearranging**: Move terms across the equation to isolate the variable of interest.
• **Simplifying Fractions**: Sometimes you need to combine or reduce fractions to make them easier to work with.
• **Balancing Equations**: Ensure any operations performed on one side of an equation are mirrored on the other to maintain equality.

In this exercise, starting from the set condition \( y=0 \), manipulating expressions and equations appropriately allowed us to determine the \( x \- ext{intercept} \) efficiently. Mastery of algebraic manipulation is essential for solving rational functions and other types of equations.