Problem 7 - College Algebra and Calculus: An Applied Approach

Question

In Exercises, evaluate the function. If necessary, use a graphing utility, rounding your answers to three decimal places. \(f(x)=2^{x-1}\) (a) \(f(3)\) (b) \(f\left(\frac{1}{2}\right)\) (c) \(f(-2)\) (d) \(f\left(-\frac{3}{2}\right)\)

Step-by-Step Solution

Answer

The values of the function f(x) at the specified points are: (a) f(3) = 4, (b) f(1/2) = 0.707, (c) f(-2) = 0.125, and (d) f(-3/2) = 0.177.

Step 1: Evaluate Function at x = 3

For the function \(f(x) = 2^{x-1}\), when \(x = 3\), plug in the value of x into the function to obtain \(f(3) = 2^{3-1} = 2^2 = 4\).

Step 2: Evaluate Function at x = 1/2

For the function \(f(x) = 2^{x-1}\), when \(x = 1/2\), plug in the value of x into the function to obtain \(f(1/2) = 2^{1/2-1} = 2^{-1/2} = 0.707\). Here, the value is rounded to three decimal places.

Step 3: Evaluate Function at x = -2

For the function \(f(x) = 2^{x-1}\), when \(x = -2\), plug in the value of x into the function to obtain \(f(-2) = 2^{-2-1} = 2^{-3} = 0.125\).

Step 4: Evaluate Function at x = -3/2

For the function \(f(x) = 2^{x-1}\), when \(x = -3/2\), plug in the value of x into the function to obtain \(f(-3/2) = 2^{-3/2-1} = 2^{-5/2} = 0.177\). Again, here the value is rounded to three decimal places.

Key Concepts

Function Evaluation and Graphing Utilities

Understanding how to evaluate functions is a fundamental skill in mathematics. Graphing utilities, such as graphing calculators or software, are invaluable tools that help visualize functions and gain deeper insights into their behavior. To evaluate a function like \( f(x) = 2^{x-1} \), you substitute the given value of \( x \) into the function. For instance, \( f(3) \) means that you replace \( x \) with 3 and calculate the result.

Using graphing utilities, you can plot \( f(x) \) to see how the function increases exponentially as \( x \) increases. These utilities often allow for the evaluation of functions at numerous points simultaneously, and grant the user the ability to zoom in or out to better understand different aspects of the function's graph. When the exact value isn't easy to compute mentally, as with \( f\left(\frac{1}{2}\right) \) or \( f(-2) \), a graphing utility can provide an approximate decimal value, which can then be further refined by rounding to the desired number of decimal places.

Exponential Functions

Exponential functions, such as \( f(x) = 2^{x-1} \), are a class of functions characterized by a constant base raised to a variable exponent. They exhibit a rapid increase or decrease, which depends on the base and the sign of the exponent. For a positive base greater than one, if the exponent increases, the function's value grows exponentially, which is visually represented as a steep upward curve on a graph.

Considering the function from our exercise, when the input value for \( x \) is positive and greater, the output is a larger positive value. When you evaluate \( f(3) \), you get a simple integer value of 4. However, when you evaluate \( f(-2) \) or \( f\left(-\frac{3}{2}\right) \), the negative exponents indicate the inverses of the function's growth, resulting in fractional output values.

Rounding Decimal Places

Rounding decimal places is a technique used to simplify numerical expressions or when precision beyond a certain point is unnecessary. This is particularly useful when dealing with the results of exponential functions, yielding non-integer values that can extend to many decimal places.

For example, the solution to \( f\left(\frac{1}{2}\right) \) yields \( 2^{-1/2} \), which is approximately 0.707106781. Rounding this to three decimal places gives us 0.707, a more manageable number for most practical purposes. Similarly, \( f\left(-\frac{3}{2}\right) \) results in \( 2^{-5/2} \), or approximately 0.176776695, which, when rounded to three decimal places, becomes 0.177. This step simplifies the values, making them easier to read and utilize in subsequent calculations.