Problem 14 - Precalculus with Calculus Previews

Question

Find an exponential function \(f(x)\) \(=b^{x}\) such that the graph of \(f\) passes through the given point. $$ (-1,5) $$

Step-by-Step Solution

Answer

The function is \(f(x) = \left(\frac{1}{5}\right)^x\).

Step 1: Recognize the given point relation

Since the point \((-1,5)\) is on the graph of the function, it satisfies the equation \(f(x) = b^x\). Hence, substituting \(x = -1\) and \(f(x) = 5\), the equation becomes \(5 = b^{-1}\).

Step 2: Convert the equation

Rewrite the equation \(5 = b^{-1}\) to find \(b\). This is equivalent to \(b^{-1} = \frac{1}{b} = 5\).

Step 3: Solve for the base \(b\)

To solve \(\frac{1}{b} = 5\), we rearrange it to find \(b\). This gives \(b = \frac{1}{5}\).

Step 4: Write the exponential function

With \(b = \frac{1}{5}\), the exponential function that fits the given point is \(f(x) = \left(\frac{1}{5}\right)^x\).

Key Concepts

Exponential Equation

An exponential equation involves the expression of a variable in the exponent. It usually takes the form \( f(x) = b^x \), where \( b \) is a positive constant. This format distinguishes exponential equations from polynomials where the variable is the base instead of the exponent. In our exercise, we dealt with an exponential equation to find a function whose graph intersects a specific point. When given a point such as \((-1,5)\), we substitute the coordinates into the exponential equation \( f(x) = b^x \). This helps us to find the value of the base \( b \) that satisfies the condition of the point being on the curve. Solving the equation involves manipulating exponents, and might require knowledge of the rules of exponents and algebraic rearrangement to isolate the base.

Graphing Exponential Functions

Graphing exponential functions involves plotting a curve that represents the equation \( f(x) = b^x \). These functions have distinctive characteristics:

• They feature continuous growth or decay.
• When the base \( b > 1 \), the function demonstrates exponential growth and slopes upward from left to right.
• For \( 0 < b < 1 \), such as \( f(x) = \left(\frac{1}{5}\right)^x \) in our exercise, the graph shows exponential decay.

Usually, the curve asymptomatically approaches the x-axis but never actually touches it. To graph, select various values of \( x \), compute corresponding \( f(x) \) values, and plot these points on the coordinate plane. The curve can be sketched smoothly through these points. The exponential function's increasing or decreasing nature depends on the base value, which determines the steepness and direction of the curve.

Solving Exponential Equations

Solving exponential equations often requires analytically manipulating the equation to isolate the base or any unknowns in the exponent. When you are given a point through which the graph must pass, as in the exercise with \((-1,5)\), substitute these coordinates into the equation to determine constraints on the exponential base. For example, from \( 5 = b^{-1} \), we rewrite to find the base \( b \) as \( b^{-1} = \frac{1}{b} = 5 \). Rearranging gives \( b = \frac{1}{5} \).

• First, understand the relation of the point to the equation.
• Substitute known values to form an equation with the base as the variable.
• Use algebraic operations to solve for the base, including inversion and rearranging terms.

Once the base is found, rewrite the function or equation with this value to solve subsequent problems. This method of solving forms a key part of understanding and manipulating exponential equations and is crucial for finding specific functions that meet given conditions.