Problem 113 - Precalculus

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 the function at \(x = -7\) is \(-58\).

Step 1: Identify the given function

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

Step 2: Substitute \(x = -7\) into the function

Substitute \(-7\) for \(x\) in the function:\[h(-7) = -\frac{1}{2}\left(\frac{1}{2}\right)^{-7} + 6\]

Step 3: Simplify the exponential term

Evaluate the exponential term:\[\left(\frac{1}{2}\right)^{-7} = \left(\frac{2}{1}\right)^{7} = 2^{7}\]Calculate \(2^{7}\):\[2^{7} = 128\]

Step 4: Substitute the evaluated exponential term

Substitute \(2^{7} = 128\) back into the equation:\[h(-7) = -\frac{1}{2} \times 128 + 6\]

Step 5: Perform the arithmetic operations

Calculate \(-\frac{1}{2} \times 128\):\[-\frac{1}{2} \times 128 = -64\]Add \(-64\) and \(6\):\[h(-7) = -64 + 6 = -58\]

Step 6: Finalize the result

The evaluated function at \(x = -7\) is \[h(-7) = -58\].

Key Concepts

Substitution

Substitution is a powerful mathematical technique that allows us to replace variables with specific values. This is particularly useful when we need to evaluate functions or expressions at certain points.

In the context of our exercise, substitution involves replacing the variable \(x\) in the function \(h(x) = -\frac{1}{2}\left(\frac{1}{2}\right)^{x} + 6\) with the value \(-7\). This means wherever we see \(x\) in the equation, we plug in \(-7\) instead.

This simple step transforms the function into a numerical calculation:

• Original function: \(h(x) = -\frac{1}{2}\left(\frac{1}{2}\right)^{x} + 6\)
• After substitution: \(h(-7) = -\frac{1}{2}\left(\frac{1}{2}\right)^{-7} + 6\)

Substitution helps us move from abstract, variable-based expressions to concrete numerical problems that can be directly calculated.

Exponentiation

Exponentiation is the mathematical operation that involves raising a number, known as the base, to a certain power, known as the exponent. This process multiplies the base by itself as many times as indicated by the exponent.

In our exercise, exponentiation is applied to \(\left(\frac{1}{2}\right)^{-7}\). Here, the base is \(\frac{1}{2}\) and the exponent is \(-7\).

A negative exponent signifies that we take the reciprocal of the base and then apply the positive exponent.

• Convert the base to its reciprocal: \(\left(\frac{1}{2}\right)^{-7} = \left(2\right)^{7}\)
• Calculate \(2^{7}\): \(2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 128\)

Thus, \(\left(\frac{1}{2}\right)^{-7} = 128\). Exponentiation transforms complex multiplications into simpler arithmetic.

Arithmetic Operations

Arithmetic operations include basic mathematical computations such as addition, subtraction, multiplication, and division. These operations are essential for evaluating expressions and simplifying results.

In our exercise, after substituting and solving the exponential term, we deal with the expression \(-\frac{1}{2} \times 128 + 6 \).

Let's break down these operations:

• Multiply: \(-\frac{1}{2} \times 128\)
  The multiplication of \(-\frac{1}{2}\) by 128 results in \(-64\).
• Addition: Add \(-64\) and \(6\)
  This results in \(-58\).

These arithmetic operations simplify the substituted and exponentiated expression to find the final value, allowing us to evaluate the function: \(h(-7) = -58\). Arithmetic operations make seemingly complex expressions more accessible and solvable.