Problem 11 - Precalculus with Limits

Question

Evaluating an Exponential Function In Exercises \(7-12,\) evaluate the function at the indicated value of \(x .\) Round your result to three decimal places. $$Function$$ $$g(x)=5000\left(2^{x}\right)$$ $$Value$$ $$x=-1.5$$

Step-by-Step Solution

Answer

After calculating and rounding to three decimal places, the result of evaluating the function at \(x=-1.5\) is 1767.767.

Step 1: Identify the Problem

The task is to evaluate the function \(g(x) = 5000\cdot 2^{x}\) at \(x = -1.5\).

Step 2: Substitute the Value of x Into the Function

Substitute \(x = -1.5\) into the function to get \(g(-1.5) = 5000\cdot 2^{-1.5}\).

Step 3: Simplify the Expression Using Exponent Rules

Simplify the expression using the rule that \(a^{-n} = 1/a^{n}\). So, \(2^{-1.5} = 1/(2^{1.5})\). We can further simplify \(2^{1.5}\) as \(\sqrt{2^3} = \sqrt{8}\). Thus, \(g(-1.5) = 5000 / \sqrt{8}\).

Step 4: Compute the Numerical Answer

Calculate the numerical answer using a calculator to find the division of 5000 by \(\sqrt{8}\).

Key Concepts

Exponent Rules

Exponent rules are a set of guidelines that help simplify and perform operations involving exponents. One of the basic rules is the negative exponent rule, which states that for any nonzero number 'a' and a positive integer 'n', the expression with a negative exponent is the reciprocal of the base raised to the opposite positive exponent: \[ a^{-n} = \frac{1}{a^n} \] Understanding this rule is pivotal in evaluating exponential functions like in the example \( g(x) = 5000\cdot 2^x \) for \( x = -1.5 \). Applying this rule helps us convert \( 2^{-1.5} \) to \( \frac{1}{2^{1.5}} \). Another essential exponent rule involves fractional exponents, where a fractional exponent indicates a root. For instance, \( a^{\frac{1}{n}} \) is the n-th root of 'a'. Therefore, \( 2^{1.5} \) or \( 2^{\frac{3}{2}} \) equates to \( \sqrt{2^3} \), which simplifies our problem further.

It's important for students to commit these rules to memory as they are foundational for understanding more complex algebraic expressions and functions.

Function Evaluation

Function evaluation is the process of finding the output of a function for a particular input value. This involves substituting the input value into the function and simplifying, following operations order and algebraic rules. In our exercise, the function \( g(x) = 5000\cdot 2^x \) needs to be evaluated at \( x = -1.5 \). After substituting \( x \) with \( -1.5 \) into the function, we get \( g(-1.5) = 5000\cdot 2^{-1.5} \) which then requires simplification.

Students should take care to substitute accurately and then apply the relevant mathematical rules, such as exponent rules, to simplify the expression as much as possible before any numerical calculation. They should always check their work to ensure that every step aligns with standard algebraic practices.

Negative Exponents

Negative exponents represent the concept of division in exponential terms. The expression \( a^{-n} \) shows that we are dividing 1 by the number 'a' raised to the 'n' power. This is because a negative exponent inverts the base as part of a fraction. For \( g(-1.5) = 5000\cdot 2^{-1.5} \), we use this concept to understand that \( 2^{-1.5} \) represents \( \frac{1}{2^{1.5}} \) or \( \frac{1}{\sqrt{2^3}} \).

Grasping the concept of negative exponents is vital because it not only simplifies calculation but also helps in understanding the nature of exponential decay in functions. When approaching problems with negative exponents, remember that they signify a 'reciprocal' action, which can drastically change the value of an expression.

Numerical Calculation

Numerical calculation is the final step of evaluating an expression or function once it has been simplified as much as possible using algebraic rules. In our example, after applying exponent rules and simplifications, the expression \( g(-1.5) \) becomes \( 5000 / \sqrt{8} \). To compute this, you would typically use a calculator to find the square root of 8, and then divide 5000 by this result.

Accuracy is key in numerical calculation, so it's important to use precise tools and methods, such as a scientific calculator or appropriate software, to ensure the answer is correct to a given degree of precision. For the problem at hand, the answer needs to be rounded to three decimal places, highlighting the importance of understanding how to round numbers correctly after a calculation.