Problem 69 - Precalculus

Question

Determine whether each statement is true or false. The function \(f(x)=-e^{-x}\) has the \(y\) -intercept (0,1)

Step-by-Step Solution

Answer

False.

Step 1: Understanding the y-intercept

The y-intercept of a function is the point where the curve crosses the y-axis. This happens when the input variable, typically denoted as \(x\), is zero. Therefore, to find the y-intercept of any function, we evaluate the function at \(x=0\).

Step 2: Evaluating the function at x=0

We have the function \(f(x) = -e^{-x}\). To find the y-intercept, we substitute \(x=0\) into this equation: \(f(0) = -e^{0}\).

Step 3: Simplifying the expression

The expression \(e^{0}\) equals 1, because any number raised to the power of zero is 1. Therefore, \(-e^{0} = -1\). This means that \(f(0) = -1\).

Step 4: Concluding the y-intercept

Since \(f(0) = -1\), the coordinates of the y-intercept are \((0, -1)\). Compare this to the given point (0,1). Since \( (0,-1) eq (0,1) \), the statement is false.

Key Concepts

Exponential Function

An exponential function is a mathematical expression where the variable is located in the exponent. This type of function is commonly written in the form \( f(x) = a \cdot e^{bx} \), where \( a \) is a constant, \( e \) is the base of natural logarithms (approximately 2.718), and \( b \) is the rate of growth or decay. The beauty of exponential functions lies in their consistent multiplicative growth or decay patterns.

Exponential functions are vital because they model a variety of real-world phenomena, such as population growth, radioactive decay, and interest calculations. The function \( f(x) = -e^{-x} \) is a specific type of exponential function where the base \( e \) is raised to a negative exponent, leading to a decay function since \( b = -1 \). This negative exponent signifies a flip across the y-axis, as well as a steady decline in value as \( x \) increases.

Function Evaluation

Function evaluation is essential when you want to find the value of a function for a specific input. Simply put, it involves substituting the input value into the function and simplifying the resulting expression.

In our example, to find the y-intercept of the function \( f(x) = -e^{-x} \), we set \( x = 0 \) because the y-intercept is the point where the graph of the function crosses the y-axis. By substituting 0 for \( x \), the function simplifies:

• \( f(0) = -e^{0} \)
• \( e^{0} = 1 \) because any number to the zero power is 1.
• Therefore, \( f(0) = -1 \).

This shows us that the y-intercept of the function is actually \((0, -1)\), contrary to the given statement that it is \((0, 1)\). Being able to evaluate a function correctly helps verify crucial points about the graph and its behavior.

Graphing

Graphing functions is a crucial skill in mathematics, allowing us to visually interpret how a function behaves. For exponential functions like \( f(x) = -e^{-x} \), graphing can reveal trends such as growth or decay.

Firstly, it's important to understand the general shape of a graph for exponential functions. For \( f(x) = e^{x} \), the graph would show exponential growth. However, with \( f(x) = -e^{-x} \), two transformations occur:

• The negative sign in \( -e^{-x} \) causes a reflection across the x-axis.
• The negative exponent \( -x \) inverts the function, demonstrating decay rather than growth.

The graph will start from a point just below the x-axis at the y-intercept \((0, -1)\) and approach the x-axis as \( x \) increases, but will never cross or touch it. Observing these trends through graphing helps in understanding how different terms in the exponential function influence its overall behavior.