Problem 20 - College Algebra and Calculus: An Applied Approach

Question

Compare the graph of \(f(x)=1 / x\) with the graph of \(g\). $$g(x)=f(x+1)=\frac{1}{x+1}$$

Step-by-Step Solution

Answer

Function \(g(x)\) is the graph of function \(f(x)\) which has been shifted one unit to the left. This results in a new vertical asymptote at x=-1, and the graph now passes through the points (0,1) and (-2,-1).

Step 1: Identify function properties

First, let's understand how function f(x) behaves. It is a reciprocal function, meaning it has a hyperbola shape. The graph passes through the points (1,1) and (-1,-1) and has vertical asymptotes at x=0 and horizontal asymptotes at y=0.

Step 2: Transform function

When transforming to \(g(x)=1 / (x+1)\), the +1 in the denominator creates a horizontal shift. This means that the whole graph of f(x) is moved one unit to the left.

Step 3: Identify transformed function properties

The graph of \(g(x)=1 / (x+1)\) behaves similarly to \(f(x)\), but it is moved one unit to the left. So it passes through the points (0,1) and (-2,-1), with a vertical asymptote at x=-1 and a horizontal asymptote at y=0.

Key Concepts

Hyperbola Graph

Understanding the hyperbola graph is vital when exploring reciprocal functions such as \(f(x) = \frac{1}{x}\). A hyperbola is a type of smooth curve lying in a plane, defined as the set of all points such that the absolute difference of the distances from two fixed points (the foci) is constant. For reciprocal functions, the hyperbola graph typically consists of two separate curves or 'branches', located in opposite quadrants of the Cartesian coordinate system.

Each branch bends closer to the axes but never touches them, creating what we call 'asymptotes'. As the value of \(x\) grows larger in the positive or negative direction, the \(y\) value of \(f(x)\) gets closer and closer to zero, which explains why the horizontal asymptote is at \(y=0\). Similarly, as we approach \(x=0\), the function shoots up towards infinity, hence the vertical asymptote. The graph is symmetric about the origin, which implies that it has reflectional symmetry in the line \(y = -x\) as well as \(y = x\).

Horizontal Shift

A horizontal shift of a graph is a type of graph transformation that moves the entire graph of a function to the left or the right. This is also known as a translation. In our example, the function \(g(x) = \frac{1}{x+1}\) shows a horizontal shift of the reciprocal function \(f(x) = \frac{1}{x}\).

The '+1' within the denominator indicates that every point on the graph of \(f(x)\) is translated one unit to the left to create the graph of \(g(x)\). To visualize this, if you take any point \((a, b)\) on the original graph, the corresponding point on the shifted graph would be \((a-1, b)\). This shifting process does not affect the shape of the graph; it simply repositions it within the coordinate plane.

Vertical Asymptote

The vertical asymptote of a graph is a line that the graph approaches but never crosses or touches. The function's value gets increasingly large (positive or negative) as it gets closer to the line, but the function never reaches the line itself.

For the original function \(f(x) = \frac{1}{x}\), the vertical asymptote is at \(x = 0\). When it comes to the function \(g(x)=\frac{1}{x+1}\), we observe a horizontal shift that moves the vertical asymptote one unit to the left, placing it at \(x = -1\). This subtle change can significantly impact the domain of the function, as the domain now excludes \(x = -1\) rather than \(x = 0\), where the function is undefined.

Horizontal Asymptote

A horizontal asymptote on a graph represents a horizontal line that the graph gets closer and closer to as the value of \(x\) heads towards positive or negative infinity. For the reciprocal function \(f(x) = \frac{1}{x}\), despite any transformations that involve horizontal shifting or reflection, the horizontal asymptote remains unchanged at \(y = 0\).

This horizontal line indicates the value that \(f(x)\) or \(g(x)\) is approaching but will never actually meet as the absolute value of \(x\) becomes larger. Understanding horizontal asymptotes helps us to predict the behavior of functions at the extreme ends of the x-axis and gives a clear picture of the long-term trend of the function's graph.

Graph Transformation

Graph transformation encompasses various operations that alter the appearance of a graph. In the context of our functions \(f(x)\) and \(g(x)\), transformation includes shifting, stretching, compressing, and reflecting.

The horizontal shift is just one example of transformation. Others include vertical shifts (up or down), reflections across either axis, and stretching or compressing the graph horizontally or vertically. These transformations allow us to map one function onto another, maintaining the overall shape but altering its position or size. When comparing \(f(x)\) and \(g(x)\), it's pivotal to note that, besides the horizontal shift, there are no other transformations affecting the steepness or orientation of the hyperbolic branches.