Problem 14 - Calculus of a Single Variable

Question

In Exercises 9–16, sketch the graph of the function and state its domain. $$ f(x)=\ln x-4 $$

Step-by-Step Solution

Answer

The graph of the function \(f(x) = \ln x - 4\) is a downward shift by 4 units of the base logarithm graph. The domain of the function is \((0, \infty)\).

Step 1: Identify the Basic Form

The given function \(f(x) = \ln x - 4\) is in the form of \(y = ln(x) + k\), where k is the vertical shift of the graph. Here, there is a vertical downward shift of 4 units.

Step 2: Identify the Domain of the Function

The natural logarithm log (ln) is only defined for \(x > 0\). Thus, the domain of \(f(x)\) is all \(x\) for which \(x > 0\), or in interval notation, the domain = \((0, \infty)\).

Step 3: Sketch the Graph

The base graph of \(ln(x)\) starts from \((-∞, 0)\) and increases without bound as \(x\) approaches infinity. A downward shift of the graph by 4 units will result in sketch that starts at \((-∞, -4)\), approaches zero on the y-axis as \(x\) goes to zero from the right and increases without bound as \(x\) approaches infinity.

Key Concepts

Natural Logarithm Domain

The domain of a function is the set of all possible input values (usually represented as x) which the function can accept. When it comes to natural logarithm functions like f(x) = \(ln(x)\), their domain is quite specific due to the nature of logarithms. The natural logarithm is only defined for positive real numbers, which mathematically translates to x > 0.

Therefore, for our function f(x) = \(ln(x) - 4\), we can deduce that it can only take positive values of x. In interval notation, we express the domain of f(x) as (0, \(\infty\)). Understanding the domain is crucial because it tells us where the function exists and can be graphed on the x-axis. For example, even though the y-value of the graph can go below zero due to vertical shifts, we can't plug in negative or zero x-values into the function, and thus the function won't produce any output for those x-values.

Vertical Shifts in Graphs

Vertical shifts are a transformation that move the graph of a function up or down without changing its shape. This is achieved by adding or subtracting a constant value to the entire function, and it is reflected in the function's equation. For our function f(x) = \(ln(x) - 4\), the '-4' indicates that there is a vertical shift downward by 4 units.

Imagine the original graph of \(ln(x)\), which passes through the point (1,0) since \(ln(1) = 0\). With the vertical shift, this point would move to (1,-4). Similarly, all other points on the graph will move down by 4 units. This downwards shift affects the y-intercept of the function but does not alter the domain. The graph will still be approaching infinity as x gets larger, but the entire function is essentially 'pushed' down along the y-axis.

Sketching Graphs of Functions

Sketching the graph of a function involves understanding and applying transformations to its basic shape. For logarithmic functions like our example f(x) = \(ln(x) - 4\), the base graph of \(ln(x)\) serves as the starting point. This curve starts at negative infinity on the y-axis and slowly increases as x increases, passing through (1,0) before rising more steeply.

In sketching our function's graph, we apply a vertical shift downward by 4 units, while ensuring that the curve never crosses into the negative x-axis, due to the domain constraint (0, \(\infty\)). The resulting sketch should accurately depict the curve's approach to the y-axis as x approaches zero from the right, and its increase without bound as x gets larger, maintaining the downward shift throughout. Accurately sketching a graph not only helps in visualizing the function's behavior but also establishes a deeper understanding of the function’s properties and how they relate to its algebraic expression.