Problem 79 - Precalculus

Question

Graph the logarithmic function using transformation techniques. State the domain and range of \(f\). $$f(x)=-\log (x)+1$$

Step-by-Step Solution

Answer

Domain: \((0, \infty)\); Range: \((-\infty, 1)\).

Step 1: Understand the basic function

We start by identifying the basic format of the logarithmic function, which is \( y = \log(x) \). This is our parent function.

Step 2: Identify the transformations

The function given is \( f(x) = -\log(x) + 1 \). This means that we have two transformations:1. A vertical reflection due to the negative sign in front of \( \log(x) \).2. A vertical shift upwards by 1 unit due to the \(+1\).

Step 3: Apply vertical reflection

Reflect the graph of the parent function \( y = \log(x) \) across the x-axis. This means every y-value becomes its negative, changing \( y = \log(x) \) to \( y = -\log(x) \).

Step 4: Apply vertical shift

Shift the graph of \( y = -\log(x) \) upward by 1 unit. This results in the final graph of the function \( f(x) = -\log(x) + 1 \).

Step 5: State the domain

The domain of \( f(x) = -\log(x) + 1 \) is the set of all x-values for which the function is defined. Since \( \log(x) \) is defined for \( x > 0 \), the domain is \( (0, \infty) \).

Step 6: State the range

The range of the transformed function \( f(x) = -\log(x) + 1 \) is the set of all possible y-values. Since the original range of \( \log(x) \) is \( (-\infty, \infty) \) and it gets reflected (+1), the range becomes \( (-\infty, 1) \).

Key Concepts

Transformation Techniques

Transformation techniques in mathematics allow you to change the appearance of a graph through different types of modifications without altering its fundamental nature. In the function \( f(x) = -\log(x) + 1 \), there are two main transformations applied:

• Vertical Reflection: The negative sign in front of \( \log(x) \) indicates a vertical reflection. This means that all the y-values of the parent function \( y = \log(x) \) are flipped over the x-axis. So, if you had a point at (1, 0), after reflection it would be at (1, 0) because it lies on the x-axis. However, a point (1,2) would move to (1,-2).
• Vertical Shift: The "+1" at the end of \(-\log(x)\) shifts the whole graph upwards by 1 unit. Every point on the graph is moved one unit higher than its corresponding point in the graph of \( y = -\log(x) \).

These techniques are powerful for graphing complex equations by starting with a simpler, known graph and systematically transforming it.

Domain of a Function

The domain of a function refers to the set of all possible input values (or x-values) that will yield a valid output from the function without making any part undefined. For logarithmic functions like \( f(x) = \log(x) \), the domain is all positive real numbers because you can't take the logarithm of zero or a negative number.

In the case of \( f(x) = -\log(x) + 1 \), although transformations are applied, they do not affect the domain. Thus, the domain remains at \((0, \infty)\). This means any x-value greater than 0 is valid for this function. Understanding the domain is crucial for graphing as it tells us where the graph should begin and which areas can be excluded.

Range of a Function

The range of a function is the set of all possible output values (or y-values), which result from using the domain of the function. As the parent function \( y = \log(x) \) has a range \((-\infty, \infty)\), this reflects how small and large it moves as x increases.

With \( f(x) = -\log(x) + 1 \), the reflected range of \(-\log(x)\) would originally be \((\infty, -\infty)\), which means it decreases indefinitely. However, because we add 1 in the transformation, the highest point every possible y-value can reach is 1.

• Thus, this function's possible y-values or range become \((-\infty, 1)\), showing the major effect of the transformations on the output.

Understanding the range is key to knowing the limits of a function's output, which helps you in predicting behavior and verifying the accuracy of your graph.