Problem 63 - A Graphical Approach to Precalculus with Limits

Question

If \(f(x)=a^{x}\) and \(f(3)=27,\) find each value. (a) \(f(1)\) (b) \(f(-1)\) (c) \(f(2)\) (d) \(f(0)\)

Step-by-Step Solution

Answer

(a) 3, (b) \(\frac{1}{3}\), (c) 9, (d) 1

Step 1: Identify Known Information

We are given the function \(f(x) = a^x\) and the value \(f(3) = 27\). This means that when \(x = 3\), \(a^3 = 27\).

Step 2: Solve for the Base 'a'

To find \(a\), solve the equation \(a^3 = 27\). Since \(27 = 3^3\), it follows that \(a = 3\).

Step 3: Calculate f(1)

Now that \(a=3\), substitute \(x=1\) into the function: \(f(1) = 3^1 = 3\).

Step 4: Calculate f(-1)

Substitute \(x=-1\) into the function: \(f(-1) = 3^{-1} = \frac{1}{3}\).

Step 5: Calculate f(2)

Substitute \(x=2\) into the function: \(f(2) = 3^2 = 9\).

Step 6: Calculate f(0)

Substitute \(x=0\) into the function: \(f(0) = 3^0 = 1\).

Key Concepts

Evaluating Functions

Evaluating functions is an essential skill in mathematics that involves finding the value of a function for a given input. In this case, we have the function \( f(x) = a^x \). Evaluating a function means that we are tasked with determining the output \( f(x) \) when we plug a specific value of \( x \) into the expression for the function.

For instance, consider the example where we're asked to find \( f(1) \). Once we know the base \( a \), which in our example turns out to be 3, we can evaluate it by simply calculating \( f(1) = 3^1 = 3 \).

When evaluating functions, make sure you substitute the input value correctly in all occurrences of the variable. This step involves both understanding the function's formula and following the correct order of operations. Each step builds your understanding, and through practice, you'll become more comfortable with evaluating different kinds of functions.

Power and Exponent Rules

Power and exponent rules are foundational principles in algebra that dictate how to handle powers of numbers and expressions. Exponents signify repeated multiplication of a base number, and understanding how to manipulate them is crucial.

Few key rules include:

• Power of a Power: \((a^m)^n = a^{m \times n}\)
• Multiplying Same Bases: \(a^m \times a^n = a^{m+n}\)
• Dividing Same Bases: \(a^m \div a^n = a^{m-n}\)
• Negative Exponent: \(a^{-n} = \frac{1}{a^n}\)
• Zero Exponent: \(a^0 = 1\) (where \(a eq 0\))

Understanding these rules makes calculating powers like \(3^1\), \(3^{-1}\), or \(3^2\) more intuitive. For the problem given:
- \(3^1\) uses the identity property, resulting in 3.
- \(3^{-1}\) demonstrates the negative exponent rule, which becomes \(\frac{1}{3}\).
- \(3^0\) uses the zero exponent rule, giving us a result of 1.
These rules are vital not only in academic exercises but also in advanced computations, as they help simplify expressions and solve equations more efficiently.

Solving Equations

Solving equations involves finding the value of the variable that makes the equation true. In the context of exponential equations like \(a^3 = 27\), solving such an equation requires a keen understanding of powers and roots.

Here, we need to find the base \(a\) when raised to the power of 3 equals 27. Recognizing that \(27 = 3^3\), we discern that \(a = 3\).

To solve such equations:

• Express one side as a power if possible. This often involves recognizing potent numbers (like 8, 27, 64) as they can be expressed as powers of smaller integers.
• Compare the exponents and solve for the base if the bases and their exponents on both sides are equal.
• Use roots if an equation is not immediately clear.

In essence, solving equations, especially involving exponentials, combines several skills: breaking down numbers into their power components and applying algebraic manipulation to isolate the variable. By practicing these steps, problem-solving becomes logical and more approachable.