Question

In Exercises \(39-42,\) draw the graph and determine the domain and range of the function. $$y=\log _{3}(x-4)$$

Short answer

The domain of the function \(y=\log_{3}(x-4)\) is \(x \in (4, \infty)\) and its range is \(y \in (-\infty, \infty)\). The function has a vertical asymptote at x=4. The graph of the function is a logarithmic curve starting at the point (4,0) and extending towards positive infinite y as x increases. The graph is also tending towards x=4 from the right as y decreases but never reaches x=4.

Step-by-step solution

Understand the given function

We are dealing with a logarithmic function here, specifically \(y=\log_{3}(x-4)\). In a basic logarithmic function represented as \(y=\log_{a}(x)\), x always has to be positive. For our function, this means the inside of the log, \(x-4\), must be positive. This is translated to x being greater than 4. This will guide us in getting the domain of the function.

Define the domain of the function

The domain of a function is the possible x-values that we can put into the function. In a log function, x, or the inside of the log, must be positive. Looking at our function again, \(y=\log_{3}(x-4)\), (x-4) must be greater than 0. By solving for x, we can thus determine our domain. In this case, we have \(x > 4\). So our domain would be \(x \in (4, \infty)\).

Define the range of the function

The range of a function are the possible y values we can get after substituting all possible x-values in the function. For log functions, there is no restriction on the y-value. Therefore, the range of our function is all real numbers, denoted by \(y \in (-\infty, \infty)\).

Plotting the graph of the function

Plotting the graph of \(y=\log_{3}(x-4)\) is relatively straightforward for students who understand how to graph transformations of functions. Ordinarily, the parent function, \(y = \log_{3}(x)\), would start at x=1 at the origin. However, the (x-4) phrase shifts the graph 4 units to the right. Therefore, the graph starts from (4,0) instead of the origin (0,0). The graph would extend beyond x=4, shooting up indefinitely as x continues to increase. The graph also gets infinitely closer to the x-axis as x approaches 4 from the right, but never actually touches x=4, forming a vertical asymptote.

Key concepts

Domain of a Function

When addressing the domain of a function, we are essentially identifying which input values (x-values) can be legitimately plugged into the function without causing mathematical issues such as division by zero or taking the logarithm of a non-positive number. For the logarithmic function in our exercise, the domain is all values that make the expression inside the log, in this case, \(x-4\), greater than zero.

Thus, for \(y=\log_{3}(x-4)\), the function is defined only for \(x>4\). To picture this, imagine sliding along the number line: every position beyond 4 is part of the domain, capturing all the possibilities for x. As soon as we hit 4 or anything less, the logarithm doesn't make sense anymore; hence, those numbers are excluded. The domain is written as \(x \in (4, \infty)\), highlighting that we can choose any value greater than 4.

Range of a Function

Contrary to the domain, the range of a function is concerned with the output values (y-values) after applying the function to the entirety of its domain. For logarithmic functions like \(y=\log_{3}(x-4)\), the output can swing to any real number. This means, for every positive x we input (remember, x has to be greater than 4), we will get some real number y.

Translating this into formal language, the range of our logarithmic function is \(y \in (-\infty, \infty)\), suggesting there's no cap on how high or low the y-values can go. No matter how large or close to zero (but not zero itself) our input, x, is, logarithms can produce an entire spectrum of y-values, from the depths of negative infinity to the peaks of positive infinity.

Logarithmic Function Transformations

Understanding how to graph transformations of a function is key to accurately representing functions like \(y=\log_{3}(x-4)\). A transformation involves shifting, stretching, compressing, or reflecting the original, or 'parent', function. In our case, the function has been shifted horizontally.

To illustrate, the parent function, \(y=\log_{3}(x)\), would typically start at the point (1,0). However, the appearance of \(x-4\) inside the log means we've shifted that starting point 4 units to the right, placing the new initial point at (5,0). Understanding these transformations helps students build an accurate representation of the function's graph in their minds, and on paper, while showcasing the effect each algebraic adjustment has on the original graph.

Vertical Asymptote

A vertical asymptote is a straight line that a function approaches but never actually touches or crosses. It represents a boundary of sorts for the function's graph. For the logarithmic function \(y=\log_{3}(x-4)\), the vertical asymptote occurs at \(x=4\).

What does this mean? As the values of x near 4 from the right side, the values of y decrease without bound, dropping off toward negative infinity. This is because the logarithm grows large and negative as its argument approaches zero. However, the function will never reach x=4; it's a limit, a frontier the graph of the function will eternally approach but never cross. This quirk is essential in the graphing of logarithms and must be denoted clearly to avoid confusion.