Problem 46 - Calculus

Question

If \(f(t)=t-1\) what are \(2 f(3 t)\) and \(f(1-t)\) and \(f(t-1) ?\)

Step-by-Step Solution

Answer

\(2f(3t) = 6t - 2\), \(f(1-t) = -t\), \(f(t-1) = t-2\).

Step 1: Evaluate f(t) for basic expression

First, identify the given function:\[f(t) = t - 1\]This represents any input \(t\) to the function \(f\), which is transformed by subtracting 1.

Step 2: Evaluate f(3t) and Calculate 2f(3t)

Now, substitute \(3t\) into \(f(t)\):\[f(3t) = 3t - 1\]Next, calculate \(2f(3t)\):\[2f(3t) = 2(3t - 1) = 6t - 2\]

Step 3: Evaluate f(1-t)

Substitute \(1-t\) into \(f(t)\):\[f(1-t) = (1-t) - 1 = -t\]

Step 4: Evaluate f(t-1)

Substitute \(t-1\) into \(f(t)\):\[f(t-1) = (t-1) - 1 = t-2\]

Key Concepts

Substitution

Substitution in mathematics involves replacing a variable in an equation or function with another expression. This allows us to see how the function behaves with different inputs. When working with functions, substitution is a powerful tool to explore transformations.

For example, the function given in the exercise is \(f(t) = t - 1\). To evaluate \(f(3t)\), we use substitution by replacing every occurrence of \(t\) with \(3t\). This shows us the impact of changing our input. By substituting \(3t\) into the function, we get \(f(3t) = 3t - 1\).

Using substitution is helpful for visualizing how functions behave under different conditions. This technique is not only applicable in solving equations but also valuable for understanding various transformation applications in mathematics.

Linear Functions

Linear functions are one of the simplest types of functions, represented by the equation \(f(x) = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. They form a straight line on a graph and are characterized by a constant rate of change.

In our function \(f(t) = t - 1\), the structure is similar: it indicates a slope of 1 and a y-intercept of -1. This simple linear equation reflects a straight line where the output decreases by 1 for every increase in \(t\).

• The slope tells us how steep the line is.
• The y-intercept indicates where the line crosses the y-axis.

Understanding linear functions forms the basis for recognizing patterns and rates, helping to predict behavior in more complex mathematical contexts.

Function Evaluation

Function evaluation refers to the process of finding the output of a function given an input. In the exercise, this involves assessing expressions like \(f(3t)\), \(f(1-t)\), and \(f(t-1)\). The goal is to understand how changing inputs affect the result produced by the function.

For example, evaluating \(2f(3t)\) requires calculating \(f(3t)\) first, which is \(3t - 1\), and then multiplying this by 2, resulting in \(6t - 2\).

• Function evaluation is crucial for determining different outputs of a function when various inputs or transformations are applied.
• It helps in understanding the behavior of functions under different scenarios.

In computations involving functions, evaluating accurately is key to solving equations and understanding transformations. This ensures a clear grasp on how different manipulations change the initial setup.