Problem 54 - Precalculus with Calculus Previews

Question

Find the \(x\) -intercepts of the graph of the given function. $$ f(x)=-3^{2 x}+5 $$

Step-by-Step Solution

Answer

The \(x\)-intercept is approximately \(x \approx 0.735\).

Step 1: Understand the Definition

The \(x\)-intercept of a graph is the point where the graph crosses the \(x\)-axis. This is the point where the value of \(f(x)\) is zero. Therefore, we need to solve the equation \(f(x) = 0\) to find the \(x\)-intercepts.

Step 2: Set the Function Equal to Zero

Set the given function equal to zero: \(-3^{2x} + 5 = 0\). This means we need to solve for \(x\) in the equation \(-3^{2x} + 5 = 0\).

Step 3: Isolate the Exponential Term

To isolate the exponential term, subtract 5 from both sides of the equation: \(-3^{2x} = -5\).

Step 4: Solve the Exponential Equation

Divide both sides by -1 to simplify the equation: \(3^{2x} = 5\). Now we need to solve for \(x\).

Step 5: Solve for \(x\) Using Logarithms

Take the logarithm of both sides to solve for \(x\): \(\log(3^{2x}) = \log(5)\). Using the property of logarithms, \(2x \log(3) = \log(5)\).

Step 6: Isolate \(x\)

Divide both sides by \(2 \log(3)\) to isolate \(x\): \(x = \frac{\log(5)}{2 \log(3)}\).

Step 7: Calculate the Value

Use a calculator to evaluate \(x = \frac{\log(5)}{2 \log(3)}\), which gives approximately \(x \approx 0.735\). This is the \(x\)-intercept of the graph.

Key Concepts

Exponential Functions

Exponential functions are a type of mathematical function that involve an exponent, where the base number is raised to the power of the variable. In our given problem, the function is \[ f(x) = -3^{2x} + 5 \].
Exponential functions often have a constant base and a variable exponent. These functions can model many real-world scenarios, such as population growth or radioactive decay.

• Growth and Decay: If the base of the exponential function is greater than 1, the function represents growth. If the base is between 0 and 1, it represents decay.
• Transformations: Exponential functions can be transformed by altering the base or by adding or subtracting numbers, affecting the graph's shape or position.

The function \[ f(x) = -3^{2x} + 5 \] is a transformation where the exponent affects the growth rate, and the -3 indicates a reflection across the x-axis. The plus 5 translates the entire graph upward by 5 units.

Logarithms

Logarithms are the opposite, or inverse, of exponentiation. They answer the question: "What power do we need to raise a certain base to get a given number?"
A logarithm can be written in the form \[ ext{log}_b(a) = c \], which means \( b^c = a \).
In the step-by-step solution, logarithms are used to solve the equation \[ 3^{2x} = 5 \]. By taking the logarithm of both sides, we convert the exponential equation into a linear form that is easier to solve.

• Properties of Logarithms: Using \( ext{log}(b^c) = c ext{log}(b) \), we can simplify the problem. This property helps to extract the variable from the exponent.
• Solving Exponential Equations: By applying logarithms, you simplify the equation, allowing isolation of the variable. This is crucial for solving exponential equations.

In our example, after logging both sides, we apply properties to solve for \( x \) by recognizing that \( 2x \text{log}(3) = \text{log}(5) \) ultimately leads to \[ x = \frac{\text{log}(5)}{2 \text{log}(3)} \].

Graphing Functions

Graphing functions is a visual way of understanding how functions behave and where they intersect the axes.
For the function \[ f(x) = -3^{2x} + 5 \], the graph is shifted and reflected because of the negative base and the constant term.

• X-intercepts: These are the values of \( x \) where \( f(x) = 0 \). The point demonstrates where the function crosses the x-axis.
• Reflections and Shifts: The negative base \( -3 \) causes the graph to reflect across the x-axis. Adding 5 shifts the graph vertically upwards by 5 units.

This understanding of reflections, shifts, and intercepts assists in sketching the graph to visualize its behavior.
A graph provides insight into the function's growth, decline, and where it may level off, particularly around intercepts.