Problem 65 - Algebra for College Students

Question

Graph \(f(x)=x, f(x)=10^{x}\), and \(f(x)=\log x\) on the same set of axes.

Step-by-Step Solution

Answer

Plot linear, exponential, and logarithmic functions on one graph.

Step 1: Understand the functions

We need to graph three functions: the identity function \(f(x) = x\), the exponential function \(g(x) = 10^x\), and the logarithmic function \(h(x) = \log x\). Each of these has distinct characteristics that we'll explore and plot.

Step 2: Plot the identity function

The identity function \(f(x) = x\) is a straight line passing through the origin (0,0) with a slope of 1. It increases linearly across the graph. To plot it, choose points such as (0,0), (1,1), (2,2), and connect them with a straight line.

Step 3: Plot the exponential function

The exponential function \(g(x) = 10^x\) is a curve that starts near zero for negative x-values, passes through (0,1), and rises steeply as x increases. Plot points such as (-1,0.1), (0,1), (1,10), and (2,100), and draw a smooth curve through these points.

Step 4: Plot the logarithmic function

The logarithmic function \(h(x) = \log x\) is undefined for zero and negative x-values, starts from \((1, 0)\), and increases slowly. Plot points such as (1,0), (10,1), (100,2), and draw a smooth curve passing through them that increases slowly compared to the exponential function.

Step 5: Draw all functions on the same axes

Now, on the same set of axes, plot the three functions: the straight line \(f(x) = x\), the steep curve \(g(x) = 10^x\), and the slowly increasing curve \(h(x) = \log x\). Ensure the graph is clearly labeled, with different line styles or colors if possible for each function for easy distinction.

Key Concepts

Identity Function

The identity function is one of the simplest functions you will encounter. It is expressed as \(f(x) = x\), meaning that the output is exactly the same as the input value. This results in a straight line when graphed:

• The line passes through the origin, which is the point (0, 0).
• The slope of the line is 1, ensuring that for every unit increase in x, y increases by the same amount.

The identity function graph is perfect for understanding linear relationships. Think of it as a mirror for value comparison, since each horizontal move is matched by an equal vertical move. Whenever you need a reference point or want to understand linear growth, the identity function is your go-to example.

Exponential Function

The exponential function, expressed as \(g(x) = 10^x\), represents a different type of growth. Instead of increasing at a constant rate, an exponential function grows by a constant percentage:

• For negative x-values, the graph approaches zero but never quite touches it, showcasing how exponential decay behaves.
• At x = 0, \(g(x)\) equals 1, a crucial reference point marking where all exponential graphs cross the y-axis.
• As x increases, the function's rate of growth accelerates dramatically, turning sharply upwards.

This type of function is common in real-world scenarios involving compounded growth, such as population increases and interest rates. Understanding its unique characteristics helps in analyzing trends that expand rapidly over time.

Logarithmic Function

The logarithmic function, \(h(x) = \log x\), can be considered the inverse of an exponential function. It plots a slow growth path compared to other functions on a graph:

• The function is undefined for x ≤ 0, meaning it only appears on the positive side of the x-axis.
• It begins at (1, 0), where any base logarithm of 1 is zero.
• Growth is slow to start, as log functions require exponential increases in x to result in even moderate increases in y.

Logarithmic functions can be particularly useful for analyzing datasets that span several orders of magnitude, such as earthquake magnitudes or sound intensity levels. It's an essential tool for data scientists and mathematicians dealing with expansive value ranges.