Question

Suppose that \(f(x)=3^{x}\). (a) What is \(f(4) ?\) What point is on the graph of \(f ?\) (b) If \(f(x)=\frac{1}{9},\) what is \(x ?\) What point is on the graph of \(f ?\)

Short answer

(a) \(f(4) = 81\), Point: \((4, 81)\). (b) \(x = -2\), Point: \((-2, \frac{1}{9})\).

Step-by-step solution

Substitute and Solve for f(4)

Given the function \(f(x) = 3^x\), substitute 4 for x to find \(f(4)\). \[f(4) = 3^4\] Calculate the value of \(3^4\). \[3^4 = 3 \times 3 \times 3 \times 3 = 81\] So, \(f(4) = 81\).

Identify the Point on the Graph for f(4)

The point corresponding to the value we just found is \((4, f(4))\). Since \(f(4) = 81\), the point is \((4, 81)\).

Set f(x) Equal to 1/9 and Solve for x

We need to solve for x given \(f(x) = \frac{1}{9}\). Start by setting the function equal to \(\frac{1}{9}\). \[3^x = \frac{1}{9}\] Recognize that \(\frac{1}{9}\) can be written as a power of 3. \[\frac{1}{9} = 3^{-2}\] Thus, set \(x\) equal to \(-2\). \[3^x = 3^{-2}\] So, \(x = -2\).

Identify the Point on the Graph for x = -2

The point corresponding to the value we just found is \((x, f(x))\). Since \(f(x) = \frac{1}{9}\) when \(x = -2\), the point is \((-2, \frac{1}{9})\).

Key concepts

Function Evaluation

Evaluating a function means finding the output value of the function for a given input value. In this case, we have the exponential function \(f(x) = 3^x\). Let's evaluate this at specific values:

• To find \(f(4)\), replace \(x\) with 4 in the function: \(f(4) = 3^4\). Simply calculate \(3^4\) by multiplying 3 four times: \(3 \times 3 \times 3 \times 3 = 81\). Hence, \(f(4) = 81\).
• This value tells us that when the input is 4, the output is 81. So, we can say that the point \((4, 81)\) lies on the graph of the function.

Graph Points

Graph points are simply pairs of values that lie on the graph of a function. These pairs are written as \((x, y)\), where \(x\) is the input value and \(y\) is the function's output value at \(x\). For instance:

• Given \(f(x) = 3^x\), we found that \(f(4) = 81\). So, the point \((4, 81)\) is on the graph of the function.
• For the second part of the exercise, if \(f(x) = \frac{1}{9}\), we need to find the input \(x\) that makes this true. Solving \(3^x = \frac{1}{9}\), we find \(x = -2\).
• This tells us that, when \(x = -2\), the output is \(\frac{1}{9}\). So, the point \((-2, \frac{1}{9})\) is on the graph.

Solving Exponential Equations

To solve exponential equations, you often need to set the equation equal to a known value and find the input that matches this value. Let's break it down:

• Consider \(f(x) = 3^x\) and set it equal to \(\frac{1}{9}\). To solve for \(x\), recognize that \(\frac{1}{9}\) can be expressed as a power of 3, specifically \(3^{-2}\).
• By matching the exponents, we get the equation \(3^x = 3^{-2}\). Thus, \(x\) must equal \(-2\).
• This shows that by rewriting the known value as a power of the base, we can easily find the value of \(x\).

Remember, solving exponential equations often involves expressing given numbers as powers of the same base and then comparing exponents. This method simplifies finding the unknown exponent.