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Chapter 3: Problem 79

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

Short Answer

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

Step by step solution

01

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.
02

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 \(+1\).
03

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 \( y = -\log(x) \).
04

Apply vertical shift

Shift the graph of \( y = -\log(x) \) upward by 1 unit. This results in the final graph of the function \( f(x) = -\log(x) + 1 \).
05

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 \( (0, \infty) \).
06

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 \( (-\infty, 1) \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

In calculus we prove that the derivative of \(f+g\) is \(f^{\prime}+g^{\prime}\) and that the derivative of \(f-g\) is \(f^{\prime}-g^{\prime} .\) It is also shown in calculus that if \(f(x)=\ln x\) then \(f^{\prime}(x)=\frac{1}{x}\) Use these properties to find the derivative of \(f(x)=\ln \frac{1}{x}\)

Use a graphing calculator to plot \(y=\ln (2 x)\) and \(y=\ln 2+\ln x .\) Are they the same graph?

Solve the logarithmic equations. Round your answers to three decimal places. $$\log (2-3 x)+\log (3-2 x)=1.5$$

Refer to the following: In calculus, to find the derivative of a function of the form \(y=k^{x}\) where \(k\) is a constant, we apply logarithmic differentiation. The first step in this process consists of writing \(y=k^{x}\) in an equivalent form using the natural logarithm. Use the properties of this section to write an equivalent form of the following implicitly defined functions. $$y=4^{x} \cdot 3^{x+1}$$

Determine whether each statement is true or false. The graphs of \(y=\log x\) and \(y=\ln x\) have the same \(x\) -intercept (1,0).
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