Problem 38 - Algebra and Trigonometry with Analytic Geometry

Question

Sketch the graph of \(f\). $$ f(x)=\frac{1}{3} \log _{3} x $$

Step-by-Step Solution

Answer

The graph of \(f(x)=\frac{1}{3} \log_3 x\) is a slowly increasing curve starting from (1,0), passing through (3,1/3), and moving towards (9,2/3).

Step 1: Identify the Function Type

The function given is a logarithmic function of the form \(f(x) = a \log_{b} x\). In this case, \(a = \frac{1}{3}\) and \(b = 3\). The base of the logarithm is \(3\).

Step 2: Determine the Domain

The domain of the function \(f(x) = \frac{1}{3} \log_{3} x\) is \(x > 0\), since logarithms are only defined for positive values of \(x\).

Step 3: Find the Y-Intercept

To find the y-intercept, set \(x = 1\).\[ f(1) = \frac{1}{3} \log_{3}(1)\]Since \(\log_{3}(1) = 0\), the y-intercept is \(f(1) = 0\). The point is \((1,0)\).

Step 4: Find Characteristic Points

Pick a few characteristic points for better understanding of the graph.- **For \(x = 3\):**\[ f(3) = \frac{1}{3} \log_{3}(3) = \frac{1}{3} \times 1 = \frac{1}{3}\]- **For \(x = 9\):**\[ f(9) = \frac{1}{3} \log_{3}(9) = \frac{1}{3} \times 2 = \frac{2}{3}\]This gives us points \((3, \frac{1}{3})\) and \((9, \frac{2}{3})\).

Step 5: Sketch the Graph

Start plotting the graph with previously determined points \((1,0)\), \((3, \frac{1}{3})\), and \((9, \frac{2}{3})\). The graph will pass through these points and approach the y-axis asymptotically, without touching or crossing it. Since the leading coefficient \(\frac{1}{3}\) is positive, the graph rises slowly as \(x\) increases.

Key Concepts

Domain of a Function

The domain of a function refers to all the possible input values (usually represented as "x") that the function can accept. For the logarithmic function given, \(f(x) = \frac{1}{3} \log_{3} x\), the domain is defined as \(x > 0\).
This is because a logarithmic function is only defined for positive values of \(x\).

• Logarithms are not defined for zero or negative numbers.
• That's why the domain excludes non-positive values and begins from just above zero to infinity.

Understanding the domain is crucial as it dictates the range of values you can use to evaluate the function without running into mathematical errors.

Characteristic Points

Characteristic points on a graph are specific points which help us understand the shape and trajectory. For the function given, identifying such points can be key to outlining the curve of the graph.

• Start with easy values like the base of the logarithm itself to find coordinates. For instance:
  • When \(x = 3\): \(f(3) = \frac{1}{3} \log_{3}(3) = \frac{1}{3}\) gives us the point \((3, \frac{1}{3})\).
  • When \(x = 9\): \(f(9) = \frac{1}{3} \log_{3}(9) = \frac{2}{3}\) gives us the point \((9, \frac{2}{3})\).
• These points help create a more precise graph and allow us to see how the function behaves for increasing x-values.

Finding characteristic points will give us a skeletal outline of the graph. This is especially useful for complex functions where intuition alone might not easily guide you through the graph's behavior.

Y-Intercept

The y-intercept is where the graph crosses the y-axis. For any function, you find this by setting \(x = 1\) because any number to the power of zero is 1, simplifying any logarithmic function in the form \(f(x) = a \log_b x\).

• In our example, we calculate: \[ f(1) = \frac{1}{3} \log_{3}(1) = \frac{1}{3} \times 0 = 0 \]
• This gives us the y-intercept point \((1,0)\).

The y-intercept is often a crucial point since it helps in sketching and validating the graph's shape, especially for functions centered around symmetrical axes. It establishes a starting point for the graph.

Asymptote

An asymptote is a line that a graph approaches but never actually touches or intersects. In the case of logarithmic functions like \(f(x) = \frac{1}{3} \log_{3} x\), there's a vertical asymptote at \(x = 0\).

• The graph continues to approach the y-axis as x gets closer to zero, but it never actually reaches the axis.
• Asymptotes are crucial for understanding the boundaries and limits of the function's behavior.

As you sketch the graph, the awareness of an asymptote helps guide the curve's direction and ensures that it does not inadvertently cross points that are mathematically impossible according to the function's equation.