Problem 34 - Algebra 2 and Trigonometry

Question

In \(23-34,\) evaluate each function for the given value. Be sure to show your work. $$ \mathrm{f}(x)=4\left(\frac{1}{2}\right)^{-x}+3\left(\frac{1}{2}\right)^{-x} ; \mathrm{f}(3) $$

Step-by-Step Solution

Answer

The value of \( f(3) \) is 56.

Step 1: Understand the Function

The given function is \( f(x) = 4\left(\frac{1}{2}\right)^{-x} + 3\left(\frac{1}{2}\right)^{-x} \). It is a combination of two terms involving negative exponents.

Step 2: Substitute the Value of x

Substitute \( x = 3 \) into the function. This means updating the function to \( f(3) = 4\left(\frac{1}{2}\right)^{-3} + 3\left(\frac{1}{2}\right)^{-3} \).

Step 3: Simplify Negative Exponents

Recall that \( a^{-b} = \frac{1}{a^b} \). Therefore, \( \left(\frac{1}{2}\right)^{-3} = \left(\frac{2}{1}\right)^{3} = 2^3 = 8 \).

Step 4: Evaluate Each Term

Now, calculate each term: \( 4 \times 8 = 32 \) and \( 3 \times 8 = 24 \).

Step 5: Add the Results Together

Add the results from the previous step: \( 32 + 24 = 56 \).

Step 6: Final Answer

Therefore, \( f(3) = 56 \).

Key Concepts

Negative exponents

Negative exponents are an important concept in algebra. They indicate the reciprocal of a base raised to a positive exponent. In simpler terms, if you have a negative exponent, like in this case where it's \(a^{-b}\), this can be written as the reciprocal \(\frac{1}{a^b}\). For example, \(\left( \frac{1}{2} \right)^{-3}\) becomes \(\left( \frac{2}{1} \right)^{3}\).

• To solve, flip the base to its reciprocal.
• Change the sign of the exponent to positive.

This conversion transforms the problem into a much simpler multiplication of positive exponents, which we can compute with ease.

Function substitution

Function substitution is the process where you replace a variable in a function with a given number. For instance, in the function \(f(x) = 4\left(\frac{1}{2}\right)^{-x} + 3\left(\frac{1}{2}\right)^{-x}\), substituting \(`x = 3`)\) means every \(x\) is replaced by \(3\).Here's what you do:

• Identify the function given, such as \(f(x)\).
• Replace every instance of the variable (\(x\)) with the number given, such as 3, transforming the function into \(f(3)\).

This substitution simplifies the function into specific values that we can evaluate further.

Simplifying expressions

Simplifying expressions allows us to rewrite math equations in more straightforward terms. This is crucial when dealing with computations, as it makes finding results easier.Here's how you can simplify:

• Recognize like terms; here both terms share \(\left(\frac{1}{2}\right)^{-x}\).
• Apply exponent rules if necessary, like the negative exponent rule we reviewed.
• Combine terms by performing arithmetic on the coefficients (e.g., the \(4\) and \(3\) in our function).

Simplifying gives a cleaner and often smaller expression, making calculation more straightforward.

Arithmetic operations

Arithmetic operations are fundamental processes like addition, subtraction, multiplication, and division. In this problem, once we simplified the expressions involved, we focused on multiplication and addition.Here's the job of each operation:

• **Multiplication**: We multiplied coefficients with the base raised to the power, first computing \(4 \times 8 = 32\) and \(3 \times 8 = 24\).
• **Addition**: Post multiplication, the results \(32\) and \(24\) were added together resulting in the final output of \(56\).

These operations are straightforward but critical in transitioning from simplified expressions to your solution.