Problem 45 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 45-46: Sketch the graph of \(f\), and use the change of base formula to approximate the \(x\)-intercept. $$ f(x)=4^{x}-3 $$

Step-by-Step Solution

Answer

The x-intercept of the function \( f(x) = 4^x - 3 \) is approximately 0.7925.

Step 1: Understand the Function

The function given is \( f(x) = 4^x - 3 \). This is an exponential function where the base is 4. The graph of \( 4^x \) alone is an increasing curve, so \( f(x) \) will be the same curve shifted down by 3 units.

Step 2: Sketch the Graph

To sketch the graph, start with the basic shape of \( y = 4^x \), an exponentially increasing curve passing through \( (0, 1) \). Since our function is \( 4^x - 3 \), shift this entire graph downward by 3 units. The curve will approach \( y = -3 \) as \( x \) approaches negative infinity and will continue to increase past the x-axis.

Step 3: Determine the x-intercept

The x-intercept is the point where the function crosses the x-axis, which means \( f(x) = 0 \). We set the equation to zero to find this point: \[ 4^x - 3 = 0 \] Simplifying gives \[ 4^x = 3 \].

Step 4: Apply the Change of Base Formula

To solve \( 4^x = 3 \), take the logarithm of both sides. You can use any base, but let's use base 10 for easier computation: \[ \log_{10}(4^x) = \log_{10}(3) \] Apply the power rule: \[ x \cdot \log_{10}(4) = \log_{10}(3) \] Solve for \( x \): \[ x = \frac{\log_{10}(3)}{\log_{10}(4)} \]

Step 5: Approximate the x-intercept

Use a calculator to find the values: \( \log_{10}(3) \approx 0.4771 \) and \( \log_{10}(4) \approx 0.6021 \). So, \[ x \approx \frac{0.4771}{0.6021} \approx 0.7925 \]. This is the approximate x-intercept.

Key Concepts

Graphing Functions

Graphing functions can seem intimidating at first, but breaking it down can simplify the process. When dealing with exponential functions like \( f(x) = 4^x - 3 \), the graph can be an insightful tool to understand their behavior. Start by identifying the basic graph, which in this case, is \( y = 4^x \), characterized by its rapid increase as \( x \) becomes larger. This function crosses the y-axis at (0,1). However, our function \( f(x) \) is shifted downwards by 3 units because of \(-3\).

• Visualize the basic graph of \( y = 4^x \).
• Shift this graph down by 3 units since every point on \( f(x) \) is decreased by 3.
• Understand that the line will get closer and closer to \( y = -3 \) as \( x \) goes towards negative infinity, but never actually touch it.
• The graph will continue to rise and eventually cross the x-axis as \( x \) gets larger.

This visual model helps us comprehend how the function behaves across the x-values.

Change of Base Formula

The change of base formula is a useful tool in solving logarithmic equations involving exponential functions. For our given function \( f(x) = 4^x - 3 \), finding the x-intercept involves solving \( 4^x = 3 \). Using the change of base formula can be necessary because this equation doesn't simplify easily within the integers.

• The formula allows you to change the base of the logarithm from base 4 to your chosen base, usually 10 or \( e \) for convenience.
• In using base 10, the formula becomes: \( \log_{4}(x) = \frac{\log_{10}(x)}{\log_{10}(4)} \).
• This transformation makes it straightforward to solve using a calculator, thus providing an accurate approximation.

Mastering this formula equips you with the ability to handle a wide range of logarithmic problems effectively.

X-Intercepts

Finding the x-intercept of an exponential function involves determining when the function equals zero. This intercept is where the graph crosses the x-axis. For \( f(x) = 4^x - 3 \), setting the equation to zero gives us \( 4^x = 3 \).

• At this point, we are looking for the x-value where the y-value is zero.
• Simplifying the equation leads to using logarithms because \( x \) must be isolated as \( x = \log_{4}(3) \).
• Approximating this value with the change of base formula provides a more tangible understanding.
• After calculation, the approximate x-intercept for this function is \( x \approx 0.7925 \). It tells exactly where on the x-axis our exponential curve passes through.

This conclusion enhances our grasp of the function's behavior at the intercept.

Exponential Equations

Exponential equations like \( 4^x = 3 \) are equations where the exponent is the variable. These equations often involve growth or decay processes and require methods such as logarithms to solve.

• Recognize the type of exponential growth depicted in \( f(x) = 4^x - 3 \), fast increasing as we progress along the x-axis.
• Solving the equation \( 4^x = 3 \) involves taking the logarithm of both sides, simplifying the variable's isolation.
• By using logarithmic properties, specifically the power rule \( x \cdot \log(4) = \log(3) \), the exponent can be calculated.
• This demonstration of exponential equations reveals patterns within the applications of growth models.

These competencies are crucial for tackling various scientific and financial problems involving exponential relations.