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Chapter 4: Problem 35

Sketch the graph of each function. $$f(x)=2^{-x}-1$$

Short Answer

Expert verified
The graph begins at the point (0, 0), falls gradually towards \( y=-1 \) as \( x \) becomes larger, and rises sharply as \( x \) becomes negative. The asymptote of this graph is \( y=-1 \).

Step by step solution

01

Understanding the given function

The given function \(f(x)=2^{-x}-1\) is an exponential function because it has a base of 2 raised to the power of \( -x \). The '-1' is a shift transformation. It shifts the exponential function downward by one unit.
02

Determine the graph behavior

In general, the function \(2^{-x}\) shows exponential decay as the exponent becomes more negative. However, the function decreases as \( x \) increases. For \( x=0 \), the graph passes through the point (0,1), but with the '-1' shift, this point will be (0,0). The function approaches \( y=-1 \) as \( x \) goes to infinity.
03

Sketch the graph

Begin by plotting the y-intercept at (0, 0). As \( x \) increases, the graph approaches the horizontal asymptote at \( y=-1 \). As \( x \) decreases, \( y \) should rise sharply. We can also choose some values for \( x \) to put into the function to get some corresponding values for \( y \), and then we can plot these points, following the general behavior and the information in previous steps.

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Key Concepts

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

Graph Sketching
Graph sketching is an essential skill that helps visualize mathematical functions. When sketching the graph of an exponential function like \( f(x) = 2^{-x} - 1 \), it's important to note key points and behaviors.

  • Identify intercepts: For this function, when \(x = 0\), \( f(0) = 2^{0} - 1 = 0\). So, the graph passes through the point (0,0).
  • Understand asymptotic behavior: As \( x \) approaches infinity, \( f(x) \) gets closer to its horizontal asymptote at \( y = -1 \).
  • Choose additional points: Substitute values for \( x \) to estimate more points, like \( x = 1 \) giving \( f(1) = \frac{1}{2} - 1 = -\frac{1}{2} \), and plot these on the graph.
  • Consider symmetry and trends: The graph of \( f(x) = 2^{-x} - 1 \) will decay since the base of the exponent is less than 1 due to the negative power.

Plot these points and smoothly join them to form the curve. Respect the asymptotes as limits the graph never crosses.
Exponential Decay
Exponential decay occurs when a function reduces towards a baseline as the input grows. For the function \( f(x) = 2^{-x} - 1 \), the base is part of what causes this decay.

  • Exponential nature: Here, the function \( 2^{-x} \) decays because the exponent \(-x\) inversely affects the growth. As x becomes positive, \( 2^{-x} \) becomes a fraction, decreasing overall value.
  • Decay behavior: The graph tends toward its asymptote \( y = -1 \) since \( 2^{-x} + c \) approaches \( 0 + c \) as \( x \to \infty \).
  • Real-world context: This kind of decay often models things like radioactive decay or cooling temperatures, where quantities decrease rapidly at first then stabilize.

Understanding decay helps in visualizing the graph correctly to see how the function ebbs over increasing \( x \).
Transformations
Transformations modify functions to shift, stretch, or compress their graphs. For \( f(x) = 2^{-x} - 1 \), specific transformations make the graph distinctive.

  • Vertical shifts: The \(-1\) in \( f(x) = 2^{-x} - 1 \) shifts the whole graph downwards by 1 unit, altering the baseline position.
  • Horizontal transformations: The negative sign in the exponent \(-x\) flips the graph left to right compared to a typical exponential growth function.
  • Transformation impact: The combined result is a graph that decays (goes down) starting from (0,0), higher up then peels away toward \( y = -1 \) moving right.

Applying these transformations influences the graph's behavior and starting points, exemplifying how mathematical changes alter visual representations.

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Most popular questions from this chapter

Explain why the function \(f(t)=e^{(1 / 2) t}\) cannot model exponential decay.

Find the inverse of the given function. Then graph the given function and its inverse on the same set of axes. $$f(x)=\frac{x+3}{x}$$

This set of exercises will draw on the ideas presented in this section and your general math background. Explain why the equation \(2 e^{x}=-1\) has no solution.

In \(1965,\) Gordon Moore, then director of Intel research, conjectured that the number of transistors that fit on a computer chip doubles every few years. This has come to be known as Moore's Law. Analysis of data from Intel Corporation yields the following model of the number of transistors per chip over time: $$s(t)=2297.1 e^{0.3316 t}$$ where \(s(t)\) is the number of transistors per chip and \(t\) is the number of years since \(1971 .\) (Source: Intel Corporation) (a) According to this model, what was the number of transistors per chip in \(1971 ?\) (b) How long did it take for the number of transistors to double?

Solve the logarithmic equation and eliminate any extraneous solutions. If there are no solutions, so state. $$\log _{2}(x+5)=\log _{2}(x)+\log _{2}(x-3)$$
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