Problem 45 - College Algebra with Corequisite Support

Question

For the following exercises, evaluate each function. Round answers to four decimal places, if necessary. \(f(x)=-4^{2 x+3},\) for \(f(-1)\)

Step-by-Step Solution

Answer

The value of \(f(-1)\) is \(-4\).

Step 1: Substitute the Value

To find the value of the function given, substitute \(x = -1\) into the function \(f(x) = -4^{2x + 3}\). It becomes: \(f(-1) = -4^{2(-1) + 3}\)

Step 2: Simplify the Exponent

Calculate the expression in the exponent. This means we need to compute \(2(-1) + 3\): \(2(-1) + 3 = -2 + 3 = 1\).

Step 3: Calculate the Expression

Now, substitute this back into the equation:\(f(-1) = -4^1\).Calculate this expression, which is simply \(-4\).

Key Concepts

Exponents

Exponents are a way of indicating that a number, known as the base, is to be multiplied by itself a certain number of times. When you see a number like \( 4^3 \), it means that 4 should be multiplied by itself 3 times: \( 4 \times 4 \times 4 \).
In our exercise, we have an exponent of \( 2x + 3 \) applied to the base -4. When we evaluate an exponent, we're essentially working out how many times to multiply the base number. Here, since we've found \( x = -1 \), the exponent simplifies to 1.
This means we multiply -4 by itself once, which is simply

• \(-4^1 = -4 \)

Notice, exponents follow specific rules. A higher exponent means more multiplication, while an exponent of 1 means no multiplication is necessary—only the base itself is your final result.

Substitution

Substitution involves replacing a variable with a given number. It is a technique commonly used in algebra to find the value of a function or an equation, which makes problems simpler to solve.
For our given exercise, we start with the function \( f(x) = -4^{2x + 3} \). When we're asked to evaluate this function for \( x = -1 \), we substitute -1 directly into the expression wherever we see \( x \).
This transformation gives us:

• \( f(-1) = -4^{2(-1) + 3} \)

By substituting and then simplifying the expression, we can solve the problem using straightforward arithmetic, removing the variable from the equation entirely.

Simplification

Simplification is the process of reducing complex expressions into simpler or more manageable terms, making them easier to evaluate or understand.
In our problem, once substitution is done, simplification follows naturally. We reduced the exponent part of the function by calculating \(2(-1) + 3\), which equals 1.
Thus, our task boiled down to calculating:

• \(f(-1) = -4^1 = -4\)

Simplification requires applying basic arithmetic operations and looking for opportunities to consolidate parts of equations. By doing so, you break down a problem into simple, bite-sized portions that are easy to digest and solve. It's an essential skill that helps in reducing possible errors and misunderstanding during calculations.