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Chapter 5: Problem 61

Let \(f(x)=e^{x} .\) Let \(L\) denote the function that is the inverse of \(f\) (a) On the same set of axes, sketch the graphs of \(f\) and \(L\) Hint: You do not need the equation for \(L(x)\) (b) Specify the domain, range, intercept, and asymptote for the function \(L\) and its graph. (c) Graph each of the following functions. Specify the intercept and asymptote in each case. (i) \(y=-L(x)\) (ii) \(y=L(-x)\) (iii) \(y=L(x-1)\)

Short Answer

Expert verified
Sketch reflections and translates of \( \ln(x) \); \( L(x) = \ln(x) \) has domain \((0, \infty)\), range \((-\infty, \infty)\); shifts affect asymptotes and intercepts.

Step by step solution

01

Understanding Inverses

The exponential function \( f(x) = e^x \) has a basic property where it is always increasing and passes through the point \( (0,1) \). Its inverse function \( L(x) \), by definition, is the natural logarithm function \( L(x) = \ln(x) \). This is because the exponential function and the natural logarithm function are inverses of each other.
02

Sketching Graphs

Plot the exponential function \( f(x) = e^x \) which is an increasing curve passing through \( (0,1) \) and approaching the x-axis as an asymptote. For the inverse \( L(x) = \ln(x) \), plot a curve passing through \( (1,0) \), increasing slowly, and having the y-axis as an asymptote. Both graphs are reflections of each other over the line \( y = x \).
03

Discuss Domain and Range of L(x)

The domain of \( L(x) = \ln(x) \) is \( (0, \, \infty) \), while its range is \( (-\infty, \, \infty) \). This is because the logarithm function is defined for all positive x-values and can produce any real value. The graph has no y-intercept, and the x-axis serves as a horizontal asymptote.
04

Analyze y=-L(x)

For \( y = -L(x) \), simply reflect the graph of \( L(x) = \ln(x) \) over the x-axis. The domain remains \( (0, \, \infty) \), the range becomes \( (-\infty, \, 0) \), no x-intercept, and the x-axis is still the asymptote.
05

Analyze y=L(-x)

For \( y = L(-x) \), this expression results in a reflection of \( L(x) \) over the y-axis. However, L(-x) is only defined for negative \( x \), meaning no graph exists in the conventional xy-space for positive x. This function graph is not typically considered valid in the real number space.
06

Analyze y=L(x-1)

For \( y = L(x-1) \), the graph of \( L(x) \) shifts one unit to the right. The domain is now \( (1, \, \infty) \), with the range \( (-\infty, \, \infty) \). It has a vertical asymptote at \( x = 1 \) and no intercepts on the y-axis.

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Key Concepts

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

Exponential Functions
An exponential function is a mathematical function in the form of \( f(x) = a^x \), where \( a \) is a positive constant called the base. However, the most common base used in exponential functions is the Euler's number \( e \), approximately equal to 2.718.
For example, \( f(x) = e^x \) is a classic exponential function, which is continuously increasing and never touches the x-axis.

Exponential functions have several unique characteristics:
  • They show rapid growth or decay depending on whether the exponent \( x \) is positive or negative.
  • The graph of an exponential function \( f(x) = e^x \) rises sharply as \( x \) increases and approaches the x-axis but never touches it as \( x \) decreases.
  • They have a horizontal asymptote along the x-axis, meaning the curve gets infinitely close to the axis but never intersects it.
  • By its nature, \( f(x) = e^x \) does not have any x-intercepts, but crosses the y-axis at \( y = 1 \).
Graph Transformations
Graph transformations involve shifting, stretching, compressing, or reflecting the graph of an original function to obtain a new function. It's a powerful tool to visualize and analyze changes or modifications to functions.

Some common types of graph transformations include:
  • Vertical Shifts: Adding or subtracting a constant \( k \) to the function, such as \( f(x) + k \), moves the graph up or down.
  • Horizontal Shifts: Adding or subtracting a constant \( h \) inside the function, such as \( f(x - h) \), moves the graph left or right.
  • Reflecting: Multiplying the function by -1, such as \( -f(x) \), reflects it over the x-axis.
  • Stretching and Compressing: Multiplying the function by a factor greater than 1 stretches it, while multiplying by a factor between 0 and 1 compresses it.
For example, with inverse functions like \( L(x) = \ln(x) \), transformations need careful consideration due to their defined domain and range. Horizontal shifts, such as \( L(x - 1) \), move the graph rightwards, important for analyzing domain changes.
Domain and Range
In mathematics, a function's domain is the complete set of possible values of the independent variable, while the range is the complete set of possible results of the function.

Understanding domain and range is critical, particularly for inverse functions:
  • Domain of Exponential Functions: For \( f(x) = e^x \), the domain is all real numbers \( (-\infty, \infty) \) since you can substitute any value for \( x \).
  • Domain of Logarithmic Functions: For \( L(x) = \ln(x) \), the domain is strictly positive numbers \( (0, \infty) \), because logarithms of non-positive values are undefined.
  • Range of Exponential Functions: For \( f(x) = e^x \), the range is \( (0, \infty) \), since exponential functions produce only positive outputs.
  • Range of Logarithmic Functions: For \( L(x) = \ln(x) \), the range is all real numbers \( (-\infty, \infty) \), as logarithmic functions can return every real value depending on their input.
Asymptotes
Asymptotes are lines that a graph approaches but never actually touches. They are fundamental in understanding the behavior of functions, especially non-linear ones like exponential and logarithmic functions.

There are several types of asymptotes:
  • Horizontal Asymptotes: In exponential functions like \( f(x) = e^x \), the horizontal asymptote is the line \( y = 0 \) (the x-axis), as the function approaches but never reaches this line as \( x \) decreases.
  • Vertical Asymptotes: For the inverse logarithmic function \( L(x) = \ln(x) \), the vertical asymptote is the line \( x = 0 \), since the function rapidly increases as \( x \) approaches zero from the positive side.
  • Understanding asymptotes helps analyze how a function behaves at the extremes, such as very large or very small values of \( x \).
Properly identifying asymptotes is crucial when sketching graphs or transforming functions to ensure accurate representation of their behavior.

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Most popular questions from this chapter

Simplify each expression. (a) \(\ln e\) (b) \(\ln e^{-2}\) (c) \((\ln e)^{-2}\)

The Chernobyl nuclear explosion (in the former Soviet Union, on April 26,1986 ) released large amounts of radioactive substances into the atmosphere. These substances included cesium-137, iodine-131, and strontium-90. Although the radioactive material covered many countries, the actual amount and intensity of the fallout varied greatly from country to country, due to vagaries of the weather and the winds. One area that was particularly hard hit was Lapland, where heavy rainfall occurred just when the Chernobyl cloud was overhead. (a) Many of the pastures in Lapland were contaminated with cesium-137, a radioactive substance with a half- life of 33 years. If the amount of cesium- 137 was found to be ten times the normal level, how long would it take until the level returned to normal? Hint: Let \(\mathcal{N}_{0}\) be the amount that is ten times the normal level. Then you want to find the time when \(\mathcal{N}(t)=\mathcal{N}_{0} / 10\) (b) Follow part (a), but assume that the amount of cesium-137 was 100 times the normal level. Remark: Several days after the explosion, it was reported that the level of cesium- 137 in the air over Sweden was 10,000 times the normal level. Fortunately there was little or no rainfall.

Use the following information on \(p H\) Chemists define pH by the formula pH \(=-\log _{10}\left[\mathrm{H}^{+}\right],\) where [H \(^{+}\) ] is the hydrogen ion concentration measured in moles per liter. For example, if \(\left[\mathrm{H}^{+}\right]=10^{-5},\) then \(p H=5 .\) Solutions with \(a\) pH of 7 are said to be neutral; a p \(H\) below 7 indicates an acid: and a pH above 7 indicates a base. (A calculator is helpful for Exercises 49 and 50.1 An unknown substance has a hydrogen ion concentration of \(3.5 \times 10^{-9} .\) Classify the substance as acid or base.

Estimate a value for \(x\) such that \(\log _{2} x=100 .\) Use the approximation \(10^{3}=2^{10}\) to express your answer as a power of 10 Answer: \(10^{30}\)

In the text we showed that the relative growth rate for the function \(\mathcal{N}(t)=\mathcal{N}_{0} e^{k t}\) is constant for all time intervals of unit length, \([t, t+1] .\) Recall that we did this by computing the relative change \([\mathcal{N}(t+1)-\mathcal{N}(t)] / \mathcal{N}(t)\) and noting that the result was a constant, independent of \(t .\) (If you've completed the previous exercise, you've done this calculation for yourself.) Now consider a time interval of arbitrary length, \([t, t+d] .\) The relative change in the function \(\mathcal{N}(t)=\mathcal{N}_{0} e^{k t}\) over this time interval is \([\mathcal{N}(t+d)-\mathcal{N}(t)] / \mathcal{N}(t) .\) Show that this quantity is a constant, independent of \(t .\) (The expression that you obtain for the constant will contain \(e\) and \(d\), but not \(t\) As a check on your work, replace \(d\) by 1 in the expression you obtain and make sure the result is the same as that in the text where we worked with intervals of length \(d=1 .)\)
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