Problem 4 - College Algebra with Corequisite Support

Question

The graph of \(f(x)=\left(\frac{1}{2}\right)^{-x}\) is reflected about the \(y\) -axis and compressed vertically by a factor of \(\frac{1}{5} . \quad\) What is the equation of the new function, \(g(x) ?\) State its \(y\) -intercept, domain, and range.

Step-by-Step Solution

Answer

The new function is \( g(x) = \frac{1}{5} \cdot 2^x \) with y-intercept \((0, \frac{1}{5})\), domain \((-\infty, \infty)\), and range \((0, \infty)\).

Step 1: Understand the Initial Function

The original function given is \( f(x) = \left(\frac{1}{2}\right)^{-x} \), which is an exponential function. The base of the function is \( \frac{1}{2} \), and the power is \(-x\). Because of the negative sign in the exponent, this is equivalent to multiplying the base by the reciprocal when the exponent is positive, resulting in \( f(x) = 2^x \).

Step 2: Refletion about the y-axis

Reflecting a function about the y-axis means replacing \( x \) with \( -x \) in the original function. However, since the original was already formulated as \((1/2)^{-x}\), the reflection will be the same form, i.e., \( f(x) = 2^x \) remains unchanged in terms of reflection.

Step 3: Apply Vertical Compression

To vertically compress the function by a factor of \( \frac{1}{5} \), we multiply the function by \( \frac{1}{5} \). Therefore, the new function \( g(x) \) becomes: \[ g(x) = \frac{1}{5} \cdot 2^x \].

Step 4: Determine the y-intercept

The y-intercept of a function is found by setting \( x = 0 \). For \( g(x) = \frac{1}{5} \cdot 2^x \), substituting \( x = 0 \) gives \( g(0) = \frac{1}{5} \cdot 2^0 = \frac{1}{5} \cdot 1 = \frac{1}{5} \). Thus, the y-intercept is \( (0, \frac{1}{5}) \).

Step 5: Identify the Domain

The domain of an exponential function like \( g(x) = \frac{1}{5} \cdot 2^x \) is all real numbers, as you can input any real number for \( x \) and still obtain a real number output. Thus, the domain is \( (-\infty, \infty) \).

Step 6: Identify the Range

The range of \( g(x) = \frac{1}{5} \cdot 2^x \) consists of all positive real numbers since \( 2^x \) is always positive and multiplying by \( \frac{1}{5} \) scales it, but does not change its sign. Therefore, the range is \( (0, \infty) \).

Key Concepts

Exponential functions

Exponential functions are a crucial family of mathematical functions characterized by a constant base raised to a variable exponent. These functions have the general form \( f(x) = a^x \) where \( a \) is the base of the function and \( x \) represents the exponent or power.
Exponential functions are known for their rapid growth or decay, depending on the value of the base:

• If \( a > 1 \), the function exhibits exponential growth as \( x \) increases.
• If \( 0 < a < 1 \), the function shows exponential decay as \( x \) increases.

In the exercise given, the initial function is \( f(x) = \left(\frac{1}{2}\right)^{-x} \). This can be rewritten as \( f(x) = 2^x \), utilizing the property \( a^{-b} = \frac{1}{a^b} \) which demonstrates the reciprocal nature. This transformation shows an exponential growth model with a base of 2, as the negative exponent \( -x \) turns into positive \( x \) when reflected over the y-axis.

Reflection of functions

Reflecting a function over an axis gives it a mirrored appearance across that axis. When reflecting over the y-axis, you replace \( x \) in the function with \( -x \).
This reflection influences the graph: it flips the graph horizontally, changing its orientation. For functions like \( f(x) = \left(\frac{1}{2}\right)^{-x} = 2^x \), when we reflect them about the y-axis, theoretically, it becomes \( f(x) = \left(\frac{1}{2}\right)^{x} \). However, since \( f(x) = 2^x \) remained unchanged after replacing \( x \) with \( -x \), the effect has already been accounted for.
Reflections are particularly helpful for visualizing how functions behave under transformation, serving as a foundational concept in graph transformations that may later influence manipulations like compressions or stretches.

Vertical compression of functions

Vertical compression of a function changes its graph height, making it "squished down" towards the x-axis. This transformation involves taking the original function and multiplying it by a constant lesser than 1.
In the case of an exponential function like \( f(x) = 2^x \), compressing it vertically by a factor of \( \frac{1}{5} \) means multiplying it directly, resulting in the new function \( g(x) = \frac{1}{5} \cdot 2^x \).
This transformation alters several properties of the function:

• The y-intercept becomes \( (0, \frac{1}{5}) \) since \( g(0) = \frac{1}{5} \).
• The function's range stays positive but starts at a lower value closer to zero, specific to \( (0, \infty) \).

Understanding vertical compression is essential for interpreting how functions change when subjected to scaling transformations, making them an integral part of graph transformations.