Question

Is it possible to transform the graph of \(f(x)=e^x\) to obtain the graph of \(g(x)=\ln x\) ? Explain your reasoning.

Short answer

Yes, it is possible to transform the graph of \(f(x)=e^x\) to obtain the graph of \(g(x)=\ln x\) through a reflection over the line \(y=x\). This is possible because these two functions are inverse of each other.

Step-by-step solution

Understand the given functions

Firstly, you need to understand the two given functions. The function \(f(x)=e^x\) is an exponential function with base \(e\), while \(g(x)=ln x\) is a natural logarithmic function. These functions are the inverse of each other, meaning if \(f(g(x))=x\) and \(g(f(x))=x\).

Understand the transformation using the concept of inverse

The graph of \(f(x)=e^x\) and the graph of \(g(x)=ln x\) can be obtained from each other by taking a reflection over the line \(y=x\). The reflection over the line \(y=x\) transforms the point \((x, f(x))\) on the graph of \(f\) to the point \((f(x), x)\) which is on the graph of \(g = f^{-1}\), the inverse of \(f\). This is a typical transformation used when dealing with inverse functions.

Conclude the reasoning

The conclusion lies in the fact that both functions are inverses of each other. Thus, by reflecting the graph of \(f(x) = e^x\) over the line \(y = x\), one can obtain the graph of \(g(x) = ln x\). This is because the transformation in the form of reflection over \(y=x\) line will swap the initial 'inputs' and 'outputs' ie, \((x, f(x))\) becomes \((f(x), x)\).

Key concepts

Understanding Exponential and Logarithmic Functions

In the world of mathematics, exponential functions and logarithmic functions are like two sides of the same coin. To envision this, consider the function f(x) = e^x. It is an exponential function because it involves raising a constant (e - Euler's number approximately 2.71828) to a variable power (x).

The counterpart to this function is the g(x) = ln x, otherwise known as the natural logarithmic function. Logarithms are essentially the inverse operation to exponentiation. While the exponential function asks 'what is e raised to the power of x?', the logarithmic function asks 'to what power must e be raised to get x?'. These operations are fundamental in Algebra 2 and help with solving a variety of equations involving exponential growth or decay, and in understanding phenomena across diverse fields such as physics, economics, and biology.

Reflection Over the Line y=x

A pivotal concept in transforming graphs is reflection over the line y=x. Imagine drawing a straight diagonal line through a graph where every point on this line has an identical x and y coordinate. Now, if you 'mirror' any graph across this line, you achieve what is known as a reflection.

Reflection is not just a geometric flip; it is mathematically significant. When you reflect an exponential function, like f(x) = e^x, you get its inverse, which for this case is the logarithmic function g(x) = ln x. This transformation exemplifies how you can connect an original function and its inverse visually and is crucial in understanding the relationship between exponential and logarithmic functions.

Grasping Inverse Functions in Algebra 2

When you delve into inverse functions in Algebra 2, you are essentially exploring two functions that 'undo' each other. If you have a function f(x) that takes an x and gives you a y, the inverse function f^{-1}(x) takes that y and returns the original x.

In the given exercise, the inverse function of f(x) = e^x is g(x) = ln x because e^{ln x} = x and ln(e^x) = x. It's like a mathematical dance where the steps of one function are perfectly retraced backward by its partner. Each function provides a pathway to decode the other, which is a fundamental skill in solving equations and modeling scenarios in algebra and calculus.

Transformation of Graphs in Math

In mathematics, the physical alteration of a graph's appearance, such as moving, stretching, shrinking, or reflecting it, is what we refer to as a transformation of graphs. It's a tool that helps us visualize and understand the behavioral changes in the function represented by the graph.

Through transformations, complex concepts become more accessible as you can manipulate a graph to observe how each transformation affects the curve's shape and position. Concerning inverse functions, such as with the exponential function f(x) = e^x and the logarithmic function g(x) = ln x, a reflection is a type of transformation that shows how one graph is mapped onto the other, providing a visual and conceptual bridge between the two different forms of the function.