Problem 56 - College Algebra

Question

Begin by graphing \(f(x)=\log _{2} x\). Then use transformations of this graph to graph the given function. What is the vertical asymptote? Use the graphs to determine each function's domain and range. $$ h(x)=2+\log _{2} x $$

Step-by-Step Solution

Answer

The vertical asymptote for both graphs is x=0. The domain for both functions is \(x>0\). The range for the function \(f(x) = \log _{2}x\) is all real numbers, while for \(h(x)=2+\log _{2}x\), the range is all real numbers greater than 2.

Step 1: Graphing f(x)

Start by graphing the parent function \(f(x)=\log _{2}x\). This function has a vertical asymptote at x=0 and passes through point (1,0). The domain for the function is all real numbers greater than 0, and its range is all real numbers.

Step 2: Graphing h(x)

Next, graph the transformed function \(h(x)=2+\log _{2}x\). This function is a vertical shift of the parent function \(f(x)\), shifted 2 units upwards. This means every y-value of \(f(x)\) is increased by 2 in \(h(x)\). The vertical asymptote remains at x=0. The graph of \(h(x)\) passes through point (1,2). The domain for the function is all real numbers greater than 0 and its range is all real numbers greater than 2.

Step 3: Comparing f(x) and h(x)

Comparing both functions, it can be seen that the vertical shift does not affect the vertical asymptote or the domain. The vertical asymptote remains at x=0, which is the line of undefined values for the log function. The domain for both functions is \(x>0\). The difference lies in the range - while the range of f(x) is all real numbers, the range of \(h(x)\) is all real numbers greater than 2 due to the upward shift.

Key Concepts

Logarithmic Functions

Logarithmic functions, like \(f(x) = \log_2{x}\), are the inverse of exponential functions. They are instrumental in solving problems related to growth and decay, such as in finance and nature. The function \(f(x) = \log_2{x}\) represents the power to which 2 must be raised to obtain \(x\). This means if \(f(x) = 3\), then \(x = 2^3 = 8\). Understanding their basic graph and transformations allows you to solve more complex real-world problems.

• The graph passes through the point (1,0) because \(\log_2{1} = 0\).
• The graph rises slowly and continues to rise indefinitely as \(x\) increases.
• For \(0 < x < 1\), the graph will fall sharply.

Recognizing these patterns is crucial when graphing transformations and understanding how logarithmic functions behave.

Vertical Asymptote

A vertical asymptote is a line that the graph of a function approaches but never touches or crosses. For logarithmic functions like \(f(x) = \log_2{x}\), the vertical asymptote is a direct consequence of the function's domain. Because logarithmic functions are only defined for positive real numbers, there is a vertical asymptote at \(x = 0\). Here:

• As \(x\) approaches 0 from the positive side, the value of \(f(x)\) decreases without bound, meaning it heads towards negative infinity.
• The vertical asymptote is a crucial feature as it indicates where the values of the function are undefined.

This makes the asymptote essential to identify when analyzing transformation effects on the graph such as shifts and stretches.

Domain and Range

Understanding the domain and range of a function is key to grasping its behavior and solving equations. The domain of \(f(x) = \log_2{x}\) is all positive real numbers, \(x > 0\), because the logarithm of zero or negative numbers is undefined. Meanwhile, its range is all real numbers because as \(x\) increases from 0 to infinity, \(f(x)\) covers all possible real output values.

• The domain remains \(x > 0\) even when transformations are applied, like in \(h(x) = 2 + \log_2{x}\).
• The range changes with transformations: the vertical shift modifies the range from all real numbers to all real numbers greater than a specific value.

By keeping track of domain and range, you can predict and understand how different transformations affect logarithmic and other functions.

Vertical Shift

A vertical shift in graph transformations refers to moving the graph up or down. This transformation changes only the y-values of the function, affecting the range but not the domain. For instance, transforming \(f(x) = \log_2{x}\) to \(h(x) = 2 + \log_2{x}\) shifts the graph 2 units upwards.

• Every point on \(f(x)\) moves exactly 2 units up to get \(h(x)\).
• The vertical asymptote at \(x = 0\) remains unchanged.
• The range of \(h(x)\) is now all real numbers greater than 2.

Vertical shifts are straightforward transformations and fundamental in learning how to alter and adapt function graphs.