Problem 63 - A Graphical Approach to College Algebra

Question

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

Step-by-Step Solution

Answer

(a) 3, (b) 1/3, (c) 9, (d) 1.

Step 1: Determine the Base 'a'

Since it's given that \(f(x) = a^x\) and \(f(3) = 27\), we have the equation \(a^3 = 27\). To find the base \(a\), solve for \(a\):\(a = 27^{1/3} = 3\). So, \(a = 3\).

Step 2: Calculate f(1)

Now that we know \(a = 3\), we can find \(f(1)\):\(f(1) = a^1 = 3^1 = 3\).

Step 3: Calculate f(-1)

Next, to find \(f(-1)\), use the formula with exponent -1:\(f(-1) = a^{-1} = 3^{-1} = \frac{1}{3}\).

Step 4: Calculate f(2)

Now, compute \(f(2)\) using the base 3:\(f(2) = a^2 = 3^2 = 9\).

Step 5: Calculate f(0)

Finally, find \(f(0)\) since any number raised to the power of 0 is 1:\(f(0) = a^0 = 3^0 = 1\).

Key Concepts

Exponential Expression

An exponential expression involves numbers expressed in the form of a base raised to an exponent. In our example exercise, the function given is \(f(x) = a^x\), where \(a\) is the base and \(x\) is the exponent.
For exponential expressions, the base is a constant while the exponent is a variable that determines how many times the base is multiplied by itself.
In this particular problem, we needed to first find out what the base \(a\) was. We did this by using the known value \(f(3) = 27\), leading us to determine that \(a = 3\).
Using the base, you can solve any function \(f(x)\) in terms of exponential expressions, which is crucial in mathematics.

Exponents

Exponents are the powers to which the base in an exponential expression is raised. In the function \(f(x) = a^x\), the number \(x\) is the exponent. Exponents represent repeated multiplication.
A positive exponent indicates how many times the base number is used as a factor, whereas a negative exponent indicates the reciprocal.

• For example, \(3^2\) results in \(3 \times 3 = 9\).
• For a negative exponent, such as \(3^{-1}\), you would have \(\frac{1}{3}\).

The special case of an exponent being zero, such as \(3^0\), results in \(1\), regardless of the base (assuming the base is not zero).
Understanding how to work with exponents is essential when evaluating functions like these.

Function Evaluation

Function evaluation requires substituting a given value into a function to calculate the result. In the given problem, we are evaluating the function \(f(x) = 3^x\) at specific points: \(f(1)\), \(f(-1)\), \(f(2)\), and \(f(0)\).
This process involves taking the base, which we established as \(3\), and raising it to the power of each specific substitute value:

• For \(f(1)\), compute as \(3^1 = 3\).
• For \(f(-1)\), evaluate as \(3^{-1} = \frac{1}{3}\).
• \(f(2)\) becomes \(3^2 = 9\).
• Finally, \(f(0)\) results in \(3^0 = 1\).

Each calculation illustrates how different exponents modify the result of the function, emphasizing the power of exponents in exponential functions.