Problem 6 Evaluate each function. Given ... [FREE SOLUTION]

Chapter 1: Problem 6

Evaluate each function. Given \(T(x)=5,\) find a. \(T(-3)\) b. \(T(0)\) 4 c. \(T\left(\frac{2}{7}\right)\) d. \(T(3)+T(1)\) e. \(T(x+h)\) f. \(\quad T(3 k+5)\)

Short Answer

Expert verified

The answers to the function evaluations are: a. \(T(-3) = 5\), b. \(T(0) = 5\), c. \(T(\frac{2}{7}) = 5\), d. \(T(3) + T(1) = 10\), e. \(T(x+h) = 5\) and f. \(T(3k+5) = 5\).

Step by step solution

Evaluate Function at \(x = -3\)

Since \(T(x) = 5\) for all \(x\), by plugging \(x = -3\) into the function, we get \(T(-3) = 5\).

Evaluate Function at \(x = 0\)

Similarly, by plugging in \(x = 0\) into the function, we get \(T(0) = 5\).

Evaluate Function at \(x = \frac{2}{7}\)

Plugging in \(x = \frac{2}{7}\) into the function, we also get \(T(\frac{2}{7}) = 5\).

Evaluate Function at \(x = 3\) and \(x = 1\)

For \(T(3) + T(1)\), we just substitute \(x = 3\) and \(x = 1\) into the function separately, to get \(T(3) + T(1) = 5 + 5 = 10\).

Evaluate Function at \(x = x + h\)

When \(x = x + h\), the output \(T(x + h) = 5\). That's because the output for any \(x\) in \(T(x)\) is 5.

Evaluate Function at \(x = 3k + 5\)

When \(x = 3k + 5\), plugging this into the function gives \(T(3k + 5) = 5\). Again, the output doesn't change with \(x\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Function Evaluation

Function evaluation is a straightforward concept once you understand it. At its core, evaluating a function means substituting a given value into the function in place of the variable. In the given exercise, you are asked to evaluate the constant function \(T(x) = 5\) at various points:

For \(T(-3)\), you substitute \(-3\) into the function and because \(T(x) = 5\) for any \(x\), the result is simply 5.
Similarly, \(T(0)\) is evaluated as 5 by substituting 0 into the function.
The process is identical for \(T\left(\frac{2}{7}\right)\); any substitution into the constant function \(T(x)=5\) results in 5.

As you can see, with constant functions, the value of \(x\) does not affect the output of the function. This simplifies the process of function evaluation for such functions significantly.

Function Notation

Function notation is a way of representing functions in a standardized format. The notation \(T(x)\) indicates a function named \(T\) with the independent variable \(x\).
The notation not only identifies the function but also highlights the variable involved in the operation:

\(T(-3)\) clearly shows that \(-3\) is the input to function \(T\).
For expressions like \(T(3) + T(1)\), it conveys that both 3 and 1 are inputs, and their respective function outputs are combined.
For functions like \(T(x + h)\) or \(T(3k + 5)\), function notation helps to clearly understand how the inputs vary.

Using standard function notation makes it easy to communicate and detail complicated mathematical processes efficiently. It helps avoid misunderstandings and makes the operations involving functions more precise and uniform.

Precalculus Problems

Precalculus problems often involve understanding how functions behave under different scenarios. In this exercise, the focus is on a constant function, \(T(x) = 5\), which simplifies the problem dramatically.
These exercises are particularly helpful in preparing students to handle more complex function types, as they illustrate the foundational principles of functions:

Constant functions, like this one, demonstrate how a function can produce the same output for any given input.
They also highlight the importance of understanding function behavior despite changes in inputs, like \(x + h\) and \(3k + 5\).
They introduce the concept of independent variables and how they are manipulated or presented in various contexts.

This familiarity sets a groundwork for future studies where students will encounter more variables, diverse functions, and need to construct or analyze the behavior of different types of functions more deeply.

Mathematical Functions

Mathematical functions are a fundamental concept in mathematics, serving as a relationship between two sets of numbers. The function \(T(x) = 5\) is an example of a constant function, meaning it provides a single, unchanging output for any input.
Understanding mathematical functions involves knowing their different types and how they can behave:

Constant functions, like in this exercise, provide an easy introduction since their behavior is predictable and unvarying.
More generally, functions can be linear, quadratic, or even more complex like exponential or logarithmic.
Functions help model real-world phenomena and solve practical problems by establishing a connection between varying quantities.

This exercise illustrates that while some functions remain constant, other functions might require detailed understanding and solving for different outputs based on varied inputs. Recognizing these differences is crucial for solid algebraic and mathematical knowledge.

91影视

Evaluate each function. Given \(T(x)=5,\) find a. \(T(-3)\) b. \(T(0)\) 4 c. \(T\left(\frac{2}{7}\right)\) d. \(T(3)+T(1)\) e. \(T(x+h)\) f. \(\quad T(3 k+5)\)

Short Answer

Step by step solution

Evaluate Function at \(x = -3\)

Evaluate Function at \(x = 0\)

Evaluate Function at \(x = \frac{2}{7}\)

Evaluate Function at \(x = 3\) and \(x = 1\)

Evaluate Function at \(x = x + h\)

Evaluate Function at \(x = 3k + 5\)

Key Concepts

Function Evaluation

Function Notation

Precalculus Problems

Mathematical Functions

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