Problem 45 - College Algebra

Question

For the following exercises, evaluate the exponential functions for the indicated value of \(x\). $$ h(x)=-\frac{1}{2}\left(\frac{1}{2}\right)^{x}+6 \text { for } h(-7) $$

Step-by-Step Solution

Answer

The value of \( h(-7) \) is \(-58\).

Step 1: Identify the Function

The exponential function given is \( h(x) = -\frac{1}{2}\left(\frac{1}{2}\right)^{x} + 6 \). We need to evaluate this function for \( x = -7 \).

Step 2: Substitute x with -7

Replace \( x \) in the function with \( -7 \), so the expression becomes \( h(-7) = -\frac{1}{2}\left(\frac{1}{2}\right)^{-7} + 6 \).

Step 3: Simplify the Exponential Term

Using the property of exponents \( a^{-n} = \frac{1}{a^n} \), simplify \( \left(\frac{1}{2}\right)^{-7} \) to \( 2^7 \). Hence, \( h(-7) = -\frac{1}{2} \times 2^7 + 6 \).

Step 4: Calculate Power of 2

Calculate \( 2^7 = 128 \). So, the expression becomes \( h(-7) = -\frac{1}{2} \times 128 + 6 \).

Step 5: Multiply and Simplify

Multiply \(-\frac{1}{2} \times 128 = -64 \). Now our expression is \( h(-7) = -64 + 6 \).

Step 6: Final Calculation

Compute \( -64 + 6 = -58 \). Therefore, \( h(-7) = -58 \).

Key Concepts

Evaluate Functions

Understanding how to evaluate functions is crucial when working with different types of mathematical expressions, including exponential functions. Evaluating a function means calculating its output value for a specific input value. This process involves three simple steps:

• Identify the given function and understand its structure.
• Substitute the indicated input value (often denoted as \( x \)) into the function.
• Simplify and calculate the resulting expression to find the output.

In our exercise, we started with the exponential function \( h(x) = -\frac{1}{2}\left(\frac{1}{2}\right)^x + 6 \) and evaluated it for \( x = -7 \). First, we identified and wrote down the function. Next, we replaced \( x \) with -7, transforming the expression into one involving the exponential component. The final steps involved simplifying the expression with the help of properties of exponents (which we will discuss next) and basic arithmetic calculations.

Properties of Exponents

Properties of exponents are rules that help us simplify expressions containing powers or exponents. These properties are essential when dealing with exponential functions since they streamline the calculations and prevent errors. Here are a few key properties:

• Product of Powers: \( a^m \times a^n = a^{m+n} \)
• Power of a Power: \( (a^m)^n = a^{m \times n} \)
• Negative Exponent: \( a^{-n} = \frac{1}{a^n} \)

In the original exercise, we applied the negative exponent property to transform \( \left(\frac{1}{2}\right)^{-7} \) into \( 2^7 \). This conversion is fundamental for simplifying and correctly evaluating the function. Understanding these properties allows us to confidently simplify and compute expressions like those found in exponential functions.

Simplifying Expressions

Simplifying expressions refers to the process of reducing a mathematical expression to its simplest form. This often involves applying arithmetic operations and algebraic rules, like those of exponents. When simplifying expressions, the goal is to make them as straightforward as possible. Here's how you simplify exponential functions:

• Perform any applicable exponent operations, using properties of exponents as needed.
• Carry out multiplication or division to simplify further.
• Combine any like terms or constants into a singular expression.
• Finally, perform any remaining basic arithmetic to arrive at the final simplified value.

In our example, once the expression was rewritten with positive exponents, we calculated \(2^7\) to get 128. The next step involved multiplying \(-\frac{1}{2} \times 128\) to arrive at -64. The final arithmetic step, simplifying \(-64 + 6\), yielded the result \(-58\). Successfully simplifying expressions ensures accuracy and clarity, especially within exponential functions.