Problem 7 - Contemporary Precalculus

Question

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

Step-by-Step Solution

Answer

Question: Sketch the graph of the function \(f(x) = 1 - 2^{-x}\) and identify its critical points and asymptotes. Answer: The graph of the function has a y-intercept at (0, 0), no x-intercepts, and a horizontal asymptote at y = 1. The function approaches the asymptote from below as x approaches positive infinity and decreases rapidly as x approaches negative infinity.

Step 1: Write the given function

The function we are given is: $$f(x)=1-2^{-x}$$

Step 2: Find the y-intercept and x-intercept

To find the y-intercept, we set \(x=0\) and calculate the value of the function: $$f(0) = 1 - 2^{-0} = 1 - 1 = 0$$ The y-intercept is at \((0, 0)\). To find the x-intercept, we set the function equal to 0: $$0 = 1 - 2^{-x}$$ There are no real solutions for x in this case, so there are no x-intercepts.

Step 3: Find the horizontal asymptote

As \(x\) approaches positive infinity or negative infinity, let's analyze the behavior of the function: $$\lim_{x\to \infty} 1-2^{-x} = 1 - 0 = 1$$ $$\lim_{x\to -\infty} 1-2^{-x} = 1 - \infty = -\infty$$ So, there is a horizontal asymptote at \(y=1\) when \(x\to \infty\).

Step 4: Determine behavior around the asymptote

Since we know there is a horizontal asymptote at \(y=1\), let's analyze the function as \(x\) approaches positive and negative infinity: $$f(x) = 1 - 2^{-x}$$ When \(x>0\), \(2^{-x}\) is between 0 and 1, so \(1-2^{-x}\) will be between 0 and 1. Therefore, the function will approach the horizontal asymptote from below. When \(x<0\), \(2^{-x}\) becomes very large, so \(1 - 2^{-x}\) will be a large negative number. Therefore, the function decreases rapidly when \(x\) approaches negative infinity.

Step 5: Sketch the graph

Now we have enough information to sketch the graph: 1. Plot the horizontal asymptote at \(y=1\). 2. Plot the y-intercept at \((0, 0)\). 3. Sketch the graph, approaching the horizontal asymptote from below as \(x \to \infty\). 4. Sketch the graph decreasing rapidly as \(x\) approaches negative infinity. With all this information, we can create a complete sketch of the function \(f(x) = 1 - 2^{-x}\).

Key Concepts

Y-Intercept

The y-intercept of a function is the point where the graph crosses the y-axis. To find this, we evaluate the function at \(x = 0\). For the function \(f(x) = 1 - 2^{-x}\), we substitute 0 for \(x\) as follows:
\[f(0) = 1 - 2^{0} = 1 - 1 = 0\]
This means the y-intercept is located at the point \((0, 0)\).

• The y-intercept tells us about the initial value of the function when \(x = 0\).
  
• It's an essential part of plotting the function on a graph.
  

Locating the y-intercept gives us a starting point for sketching our graph, representing the point where the function lands on the y-axis.

X-Intercept

The x-intercept is the point where the graph crosses the x-axis. To find this point, set \(f(x)\) to zero and solve for \(x\) in the equation:\[0 = 1 - 2^{-x}\]Solving for \(x\), we see:
\[1 = 2^{-x}\]
To find \(x\), take the logarithm of both sides. However, in this case, such a simple real solution does not exist. Therefore, the conclusion is:

• The function \(f(x) = 1 - 2^{-x}\) does not cross the x-axis,
  
• This means there are no x-intercepts.
  

This lack of x-intercepts indicates that the function does not have real roots, and we will not see the graph touching or passing the x-axis anywhere. This is common in exponential functions where bases or transformations don't reach specific values.

Horizontal Asymptote

A horizontal asymptote is a horizontal line that the graph of a function approaches as \(x\) goes towards infinity or negative infinity. To identify the horizontal asymptote for \(f(x) = 1 - 2^{-x}\), analyze the limits:
- As \(x\) approaches positive infinity, \(2^{-x}\) approaches 0, making \(f(x)\) approach 1. Thus,\[\lim_{x\to \infty} f(x) = 1\]
- Therefore, there is a horizontal asymptote at \(y = 1\) on the right side.
- As \(x\) approaches negative infinity, \(2^{-x}\) becomes very large, so \(1 - 2^{-x}\) goes to negative infinity.

• Horizontal asymptotes help understand the behavior of the function as x becomes very large or very small.
• They signal where the graph settles or stabilizes horizontally.
• In this function, the significant horizontal asymptote is \(y = 1\), showing stabilization on the positive x-axis side.

Recognizing horizontal asymptotes is crucial in grasping the overall shape and spread of the graph in exponential functions.

Graph Behavior

Understanding graph behavior involves analyzing how the function behaves around its intercepts and asymptotes. For the function \(f(x) = 1 - 2^{-x}\), here's a breakdown of its behavior:

When \(x > 0\):

• Since \(2^{-x}\) is between 0 and 1, \(1-2^{-x}\) ranges from 0 to 1.
• The function values lie below the horizontal asymptote \(y = 1\), approaching this asymptote from below as \(x\) increases.
  

When \(x < 0\):

• The term \(2^{-x}\) becomes a large number.
• The function has values producing large negative outcomes.
• The graph decreases sharply as \(x\) tends towards negative infinity.
  

• The interplay of these behaviors gives the graph its characteristic exponential curve,
  
• It's crucial to sketch these trends when visualizing exponential functions.
  
• Understanding positive and negative behavior helps predict the graph’s movement and slope.
  

Through this behavior, you can predict where the function is increasing, decreasing, or leveling out based on the input values, completing an accurate graph sketch.