Question

In Exercises \(57-74\), sketch the graph of the equation. Look for extrema, intercepts, symmetry, and asymptotes as necessary. Use a graphing utility to verify your result. $$ y=1+\frac{1}{x} $$

Short answer

The graph of \(y=1+\frac{1}{x}\) looks like the graph of the function \(y=\frac{1}{x}\) shifted upwards by 1 unit, has an x-intercept at -1, no y-intercept, no symmetry, and has vertical asymptote at x=0 and horizontal asymptote at y=1.

Step-by-step solution

Identify the Behavior of the Function

The given function is \(y=1+\frac{1}{x}\). This is a reciprocal function which has been shifted upwards by 1 unit. Typically, the graph of a basic reciprocal function \(y=\frac{1}{x}\) will resemble two curves located in the first and third quadrants. The function will approach 0 as the value of x increases or decreases. The vertical shift of 1 means the entire graph of the function will be shifted upwards by one unit. This shift will not affect the overall shape of the graph but rather its position on the coordinate plane.

Find the Intercepts

To find the y-intercept, set x=0 in the function. However, this leads to division by zero, which is undefined. Therefore, the function does not have a y-intercept. To find the x-intercept, set the function equal to 0, that is, solve the equation \(0=1+\frac{1}{x}\) for x. Multiplying each side by x gives the equation 0=x+1. Solving for x gives x=-1. Therefore, the graph intersects the x-axis at x=-1.

Determine the Symmetry

To check for symmetry, replace x in the equation by -x. If the original equation is obtained, then the graph is symmetric with respect to y-axis and is even. If the negative of the original equation is obtained, then the graph is symmetric with respect to the origin and is odd. The function transformed to \(y=1+\frac{1}{-x}\), which is not same as the original equation or its negative. Thus, the graph of the function is neither symmetric with respect to y-axis nor origin.

Identify the Asymptotes

An asymptote to a function is a line such that the distance between the curve and the line approaches zero as they tend to infinite. For the reciprocal function, there will be two asymptotes: one vertical (x=0) and one horizontal which, due to the shift, will be y=1 instead of y=0. Neither the graph will touch nor cross these lines.

Verification using Graphing Utility

Use a graphing utility to graph the function. Ensure that it matches the description: a graph similar to a reciprocal function displaced up by one unit, without a y-intercept, with an x-intercept at x=-1, with no symmetry, and with asymptotes at x=0 and y=1.