Problem 110 - A Graphical Approach to Precalculus with Limits

Question

In general, it is not possible to find exact solutions analytically for equations that involve exponential or logarithmic functions together with polynomial, radical, and rational functions. Solve each equation using a graphical method, and express solutions to the nearest thousandth if an approximation is appropriate. $$3^{-x}=\sqrt{x+5}$$

Step-by-Step Solution

Answer

The solution is approximately \(x = -0.445\).

Step 1: Understanding the Equation

The equation given is \(3^{-x} = \sqrt{x+5}\). This equation involves an exponential function on the left side and a radical (square root) function on the right side.

Step 2: Setting Up Functions for Graphing

Define two functions from the equation: \( y_1 = 3^{-x} \) and \( y_2 = \sqrt{x+5} \). Our goal is to find the values of \(x\) where the graphs of these two functions intersect.

Step 3: Graphing the Functions

Use graphing software or calculators to plot \( y_1 = 3^{-x} \) and \( y_2 = \sqrt{x+5} \). Note that the domain for \( y_1 \) is all real numbers, but for \( y_2 \), \(x\) must be \( \geq -5 \) because of the square root of \(x+5\).

Step 4: Finding Intersection Points

Identify the points where the two graphs intersect. These intersection points correspond to the solutions of the equation \(3^{-x} = \sqrt{x+5}\).

Step 5: Approximating Solutions

Once you have identified the intersection points, read the \(x\)-coordinates from the graph. Approximating these values to the nearest thousandth gives a solution of \(x \approx -0.445\).

Key Concepts

Exponential Functions

Exponential functions are an important concept in mathematics that describe situations of rapid growth or decay. In the given equation, we have the function \( y_1 = 3^{-x} \), showcasing an exponential decay.

• **General Form**: Exponential functions can be generally expressed as \( a^x \), where \( a \) is a constant.
• **Behavior**: As \( x \) increases, \( 3^{-x} \) decreases because the negative exponent inverts the power.
• **Domains and Ranges**: The domain is all real numbers, and the range is positive real numbers but bounded between 0 and 1 for \( 3^{-x} \).

Understanding exponential functions helps in analyzing phenomena such as population growth, radioactive decay, and, as in our problem, intersections with other functions.

Radical Functions

Radical functions involve roots, and in our equation, the function \( y_2 = \sqrt{x+5} \) includes a square root.

• **Significance**: Radical functions help in expressing quantities that vary slowly.
• **Domains**: As seen from \( \sqrt{x+5} \), the domain must satisfy \( x+5 \geq 0 \), hence \( x \geq -5 \).
• **Range**: For our function, the range is all non-negative real numbers, since square roots produce non-negative results.

Plotting this function alongside exponential ones can help find their intersection points visually, as both functions behave differently across their domains.

Intersection Points

Intersection points are where two graphs meet, and they hold the key to solving the given equation graphically.

• **Determination**: By graphing both \( y_1 \) and \( y_2 \), the intersection points can be visually identified on the graph.
• **Significance**: The \( x \)-coordinates of these points are solutions to the equation \( 3^{-x} = \sqrt{x+5} \).
• **Visual Aid**: These points provide a geometric perspective, helping us see solutions that might be complex to solve analytically.

Finding these points can often be achieved using graphing software, calculators, or manually plotting points.

Numerical Approximation

Graphical methods, while effective, often result in approximations rather than precise solutions. Numerical approximation helps refine these solutions.

• **Approach**: Once the intersection is identified, numerical methods estimate \( x \)-values to required precision.
• **Precision**: In our problem, this means finding \( x \) to the nearest thousandth, necessary for practical applications.
• **Tools**: Calculators or software can assist in finely tuning the approximation from the graph.

Numerical approximation is a valuable skill, particularly in real-world problems where an exact solution is not feasible or necessary.