Question

In Exercises, evaluate the function. If necessary, use a graphing utility, rounding your answers to three decimal places. \(f(x)=3^{x+2}\) (a) \(f(-4)\) (b) \(f\left(-\frac{1}{2}\right)\) (c) \(f(2)\) (d) \(f\left(-\frac{5}{2}\right)\)

Short answer

(a) \(f(-4) = 0.111\)\n(b) \(f(-1/2) = 5.196\)\n(c) \(f(2) = 81.000\)\n(d) \(f(-5/2) = 0.577\)

Step-by-step solution

Evaluating Function for \(f(-4)\)

Substitute -4 in place of x for the function \(f(x)\), yielding \(f(-4) = 3^{(-4+2)} = 3^{-2} = 0.111\). This is in 3-decimal places rounding off.

Evaluating Function for \(f(-1/2)\)

Substitute -1/2 in place of x for the function \(f(x)\), yielding \(f(-1/2) = 3^{(-1/2+2)} = 3^{1.5} = 5.196\). This is in 3-decimal places rounding off.

Evaluating Function for \(f(2)\)

Substitute 2 in place of x for the function \(f(x)\), yielding \(f(2) = 3^{(2+2)} = 3^4 = 81.000\). This is in 3-decimal places rounding off.

Evaluating Function for \(f(-5/2)\)

Substitute -5/2 in place of x for the function \(f(x)\), yielding \(f(-5/2) = 3^{(-5/2+2)} = 3^{-.5} = 0.577\). This is in 3-decimal places rounding off.

Key concepts

Graphing Utility

When it comes to evaluating exponential functions, a graphing utility is an invaluable tool for both visualizing the function and accurately calculating values. This versatile software allows students to input an equation, such as the exponential function in our exercise, and then view the plotted graph. It's particularly useful for understanding the nature of exponential growth or decay over a range of values.

Using the graph, it becomes clearer how the function behaves as the input value, or 'x', changes. For complex expressions where manual evaluation isn't practical or for checking your work, a graphing utility can provide immediate feedback. Make sure to become familiar with basic functions like zooming in and out or tracing a point to understand how values correlate on the graph.

Function Evaluation

Function evaluation is a process where we replace the variable in a function with a specific value to find the function's output. In the case of the exponential function presented in the exercise, we follow a systematic approach. For instance, evaluating the function at a given point involves replacing the variable 'x' with the number of interest.

In our examples, for the function given by \(f(x) = 3^{x+2}\), when evaluating \(f(-4)\), we substitute '-4' for 'x'. This substitution method is the cornerstone of function evaluation and is applicable for a wide array of functions, not just exponentials. Always keep in mind the order of operations when evaluating; this will help you achieve the correct results.

Exponential Expressions

Exponential expressions involve numbers raised to a power, typically denoted as \(b^n\), where 'b' is the base and 'n' is the exponent. When 'b' is a positive number other than 1, and 'n' varies, we get an exponential function. This type of function has unique properties; it grows or decays at a rate proportional to its value.

Understanding these expressions is crucial since they appear in many scientific scenarios, like population growth or radioactive decay. To evaluate them effectively, knowledge of exponential laws, such as \(b^{-n} = 1/(b^n)\) and \(b^{m+n} = b^m * b^n\), is important. These principles have been applied in our exercise to simplify and evaluate the given function.

Rounding Decimal Places

Rounding to a specific number of decimal places is a valuable skill to have, ensuring that the numerical results are manageable and that they adhere to any required precision standards. In scientific calculations, finance, and statistics, maintaining a consistent degree of precision is often essential.

When rounding off to three decimal places, like in our exercise, you're ensuring that only three digits remain after the decimal point. The fourth digit determines whether you round up or maintain the third decimal place - if it's five or more, you round up. Always consider the context of your problem to decide the necessary degree of precision, as sometimes more or fewer decimal places may be required.