Problem 18 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to graph \(f\) and \(g\) in the same viewing window. Approximate the point of intersection of the graphs of \(f\) and \(g .\) Then solve the equation \(f(x)=g(x)\) algebraically. $$\begin{aligned}&f(x)=2^{-x+1}-3\\\&g(x)=13\end{aligned}$$

Step-by-Step Solution

Answer

The functions \(f(x)\) and \(g(x)\) intersect at the point (-3, 13).

Step 1: Graph Functions

Graph the functions \(f(x) = 2^{-x+1} - 3\) and \(g(x) = 13\). Observing the graph, estimate the x-coordinate at which they intersect.

Step 2: Set up the equation

Set the function f(x) equal to g(x) in order to find the exact point of intersection. This yields the equation \(2^{-x+1} - 3 = 13\).

Step 3: Simplify the equation

Simplify the equation by adding 3 to both sides, resulting in \(2^{-x+1} = 16\).

Step 4: Solve for x

The goal now is to solve the equation for x. As 2 to the power of 4 equals 16, we can equate -x+1 to 4. Solving for x, we get \(x = -3\).

Key Concepts

Intersection of Graphs

When examining graphs of functions, the intersection point is where two graphs meet on the coordinate plane. This point represents the solution to the equation formed by setting these two functions equal to each other. Graphing utilities like graphing calculators or apps help visualize this by plotting both functions to easily find where they intersect.

For our functions, where we have \(f(x) = 2^{-x+1} - 3\) and \(g(x) = 13\), graphing them gives a visual clue about their intersection point. You'll see these graphs intersect at a certain x-value. This x-coordinate is crucial as it helps approximate solutions before diving into algebraic methods. This visual estimation, however, needs an algebraic verification to confirm its accuracy.

Remember:

• Graph both functions.
• Check the point(s) where they meet.
• This intersection provides a solution to \(f(x) = g(x)\).

Solve Equation Algebraically

Solving equations algebraically means manipulating the equation to find the exact values for which the two expressions are equal. In this exercise, we set the expressions \(2^{-x+1} - 3 = 13\) to determine the exact intersection point found earlier graphically.

To simplify the problem:

• First, add 3 to both sides of the equation, yielding \(2^{-x+1} = 16\).
• Next, recognize that 16 is a power of 2, specifically \(2^4\).
• Set the exponents equal: \(-x+1 = 4\)

This solves to \(x = -3\). By working through this algebra step, you validate the graphical estimation and confirm the solution accurately.

Exponentials

The function involving exponentials in this exercise is a bit tricky at first glance. An exponential function, like \(f(x) = 2^{-x+1} - 3\), revolves around the power of a number—in this case, 2.

Here's how these components work:

• Exponentials grow (or decay) at a constant rate. Here, the base is 2, offering exponential decay.
• The negative exponent \(-x+1\) inversely affects this growth, squeezing the function towards zero as x increases.
• Adjustments, like subtracting 3, translate the entire graph vertically.

Deciphering exponentials involves understanding these parts: how the base number expands or contracts with different exponents and how modifications, such as additions or subtractions, impact the graph. Recognizing these aspects helps when solving real-world problems involving exponential relationships.