Question

The shape of the graph \(\ln 1 \rightarrow t\) is (A) straight Line (B) Parabolic curve (C) Hyperbole curve (D) random shape curve

Short answer

The shape of the graph $\ln 1 \rightarrow t$ is a straight line because the natural logarithm of 1 is always 0, and the function can be rewritten as $0 \rightarrow t$. This results in a horizontal line along the t-axis. Therefore, the answer is (A) straight Line.

Step-by-step solution

Recognize the function

We have the function \(\ln{1} \rightarrow t\). As mentioned before, the natural logarithm of 1 is always equal to 0. So, this function can be rewritten as \(0 \rightarrow t\).

Evaluate the function

Since \(\ln{1} = 0\), every input (\(t\)) will generate the same output, which is 0. This function can be viewed as \(y = 0 \rightarrow t\). In this equation, no matter what \(t\) is, the result will always be 0 for \(y\).

Determine the graph shape

As no matter the value of \(t\), the result is 0, the graph will be a horizontal line along the \(x\)-axis (t-axis). This means that the shape of the graph is a straight line. The correct answer is: (A) straight Line

Key concepts

Natural Logarithm

The natural logarithm, often represented as \(\ln{(x)}\), is a logarithm to the base \(e\), where \(e\) is an irrational constant approximately equal to 2.71828. This logarithm is frequently used in mathematics due to its natural occurrence in growth processes, like that of populations or radioactive decay.
The function \(\ln(x)\) is only defined for positive values of \(x\). When computing \(\ln{1}\), we use the fact that \(e^0 = 1\), meaning the natural logarithm of 1 is simply 0. This property is very useful and often remembered as one of the characteristics of the logarithm functions.

• \(\ln(1) = 0\)
• \(\ln(x)\) is only defined for \(x > 0\)

This foundational understanding is crucial when working with functions that include natural logarithms, as it dictates the behavior of graphs and outputs of the function.

Horizontal Line

A horizontal line is a graph with an equation resembling \(y = c\), where \(c\) is a constant. This means that the value of \(y\) does not change, regardless of the value of the variable on the x-axis. In simpler terms, a horizontal line remains flat as it moves along, parallel to the x-axis.

For the function discussed in the exercise, the equation \(y = 0\) forms a horizontal line on the graph. Since the output is consistently 0, regardless of \(t\), the graph of this special function will simply be a line parallel to the x-axis (or \(t\)-axis in this context).

• Equation form: \(y = c\)
• Parallel to the x-axis
• No slope (slope = 0)

Understanding this concept helps visualize constant functions and recognize the shapes of their respective graphs, which is simply a straight horizontal line in such scenarios.

Functions

A function in mathematics is a relation where each input is associated with exactly one output. It's a fundamental concept in algebra and calculus, shaping how we understand relationships and dependencies between variables.
Functions can take various forms; some may be linear, quadratic, or even non-continuous. They are represented as equations that tell us how to compute the output for any given input.

In the exercise, the function in question \(\ln(1) \to t\) simplifies to \(y = 0\) because the output is constant for all values of \(t\). This showcases one specific type of function, known as a constant function, where the output doesn't vary.

• Each input corresponds to one output
• Can be linear, quadratic, constant, etc.
• Forms the basis for graphing equations and studying their behavior

Grasping the general idea of functions allows one to predict graph shapes and understand the nature of various mathematical representations.