Question

Describe the transformation of \(f\) represented by \(g\). Then graph each function. \(f(x)=e^x, g(x)=e^x-1\)

Short answer

The function \(g(x)=e^x-1\) is a vertical shift 1 unit downwards from the function \(f(x)=e^x\). The graph of \(f(x)=e^x\) is an upward-sloping curve that passes through the point (0,1) and approaches but never touches the x-axis as \(x\) tends towards negative infinity. The graph of \(g(x)=e^x-1\) is the same curve shifted down 1 unit, passing through the point (0,0) instead of (0,1) and approaching y=-1 as \(x\) tends towards negative infinity.

Step-by-step solution

Recognize Transformation

First, it is important to know how functions transform. The function \(g(x)=f(x)-1\) represents a vertical shift of the function \(f(x)\) downwards by 1 unit. This means that every point on the graph of \(f(x)\) moves one unit down to form the graph of \(g(x)\).

Graph Function \(f(x)\)

The graph of the function \(f(x)=e^x\) is an upward-sloping curve that passes through the point (0,1) on the y-axis, and which approaches but never touches the x-axis as \(x\) tends towards negative infinity. The curve rises rapidly for increasing values of \(x\).

Graph Function \(g(x)\)

To get the graph of \(g(x)=e^x-1\), take the graph of \(f(x)=e^x\) and shift it down by one unit. This means that the curve will now pass through the point (0,0) instead of (0,1), and it will approach y=-1 (but never touch it) as \(x\) tends towards negative infinity. It will still rise rapidly for increasing values of \(x\).

Key concepts

Understanding Vertical Shifts

Transformations are vital for understanding how changes to a function's equation affect its graph. A vertical shift occurs when every point on the graph of a function is moved the same distance in the vertical direction, either upwards or downwards. In our case, given that \(g(x) = e^x - 1\), the graph of \(f(x) = e^x\) is shifted down by 1 unit. This downward shift means all the values of \(f(x) = e^x\) are reduced by 1.
This can be understood as replacing each \(f(x)\) value with \(g(x)\) value, i.e., \(g(x) = f(x) - 1\). Thus, for each point on the graph, the y-coordinate is reduced by 1 unit, effectively translating the entire graph downward while maintaining its shape.

The Basics of Graphing Functions

Graphing functions is a key mathematical skill that involves plotting the function on a coordinate plane to see its geometric representation. The function \(f(x) = e^x\) is known for its characteristic upward-sloping exponential shape.
It starts at the point (0,1), reflecting the value of \(e^0\), and extends from left to right, increasing rapidly.

• The y-intercept of the function is at the point (0,1), because \(e^0 = 1\).
• As \(x\) decreases towards negative infinity, the curve approaches but does not touch the x-axis (y = 0), due to the nature of exponential decay.
• For positive values of \(x\), the graph grows fast, reflecting the rapid increase of the exponential function.

When graphing \(g(x) = e^x - 1\), use the graph of \(f(x) = e^x\) and apply the vertical shift. By shifting the entire graph of \(f(x)\) down by 1 unit, the y-intercept is now at (0,0), and the curve approaches the horizontal line y = -1 for negative \(x\).

Exploring Exponential Functions

Exponential functions are a fundamental concept in mathematics. They are defined by the equation \(f(x) = a^x\), where \(a\) is a positive constant, typically \(e\) (approximately 2.71828), the base of natural logarithms, especially in the continuous growth models.
Exponential functions are characterized by their rapid growth rate.

• For \(f(x) = e^x\), as \(x\) increases, \(f(x)\) increases quickly to infinity.
• The function exhibits what is known as exponential growth for positive \(x\).
• As \(x\) becomes more negative, \(f(x)\) approaches zero but never actually reaches it, embodying exponential decay.

This concept of exponential behavior is crucial not only in mathematics but across various fields, such as physics, biology, and finance, where it models phenomena ranging from population growth to compound interest. Understanding the properties of exponential functions can lead to a deeper comprehension of various real-world problems.