Problem 35 - Algebra and Trigonometry Real Mathematics, Real People

Question

Evaluate the function at each specified value of the independent variable and simplify. $$f(y)=3-\sqrt{y}$$ (a) \(f(4)\) (b) \(f(0.25)\) (c) \(f\left(4 x^{2}\right)\)

Step-by-Step Solution

Answer

The simplified expressions after substituting the given values into the given function are: (a) \(f(4) = 1\), (b) \(f(0.25) = 2.5\), and (c) \(f(4x^{2}) = 3 - 2x\).

Step 1: Evaluating \(f(4)\)

To evaluate \(f(4)\), substitute \(4\) into the function. This would yield: \(f(4) = 3 - \sqrt{4}\). The square root of \(4\) is \(2\), thus the expression simplifies to \(f(4) = 3 - 2 = 1\).

Step 2: Evaluating \(f(0.25)\)

To evaluate \(f(0.25)\), substitute \(0.25\) into the function. This would yield: \(f(0.25) = 3 - \sqrt{0.25}\). The square root of \(0.25\) is \(0.5\), thus the expression simplifies to \(f(0.25) = 3 - 0.5 = 2.5\).

Step 3: Evaluating \(f(4x^{2})\)

To evaluate \(f(4x^{2})\), substitute \(4x^{2}\) into the function. This would yield: \(f(4x^{2}) = 3 - \sqrt{4x^{2}}\). The square root of \(4x^{2}\) is \(2x\), given that the square root of \(4\) is \(2\) and the square root of \(x^{2}\) is \(x\), thus the expression simplifies to \(f(4x^{2}) = 3 - 2x\).

Key Concepts

Independent Variable

An independent variable is a fundamental concept in mathematics, particularly in functions and algebra. It represents the input or the argument of the function, whose value we can control or choose. In the exercise, the independent variable is denoted as \( y \) within the function \( f(y) = 3 - \sqrt{y} \). When evaluating a function, we substitute specific values for the independent variable to find the corresponding output of the function.

Understanding how to handle an independent variable allows one to explore various outputs of a function by varying the input. This is a critical skill in many areas of mathematics and science, as it helps to analyze and predict behaviors in models. For instance, in the exercise, you're asked to evaluate the function \( f(y) \) with different substitutions: \( 4 \), \( 0.25 \), and \( 4x^2 \).

• In \( f(4) \), the independent variable \( y \) is replaced by \( 4 \).
• In \( f(0.25) \), it becomes \( 0.25 \).
• In \( f(4x^2) \), it is replaced with \( 4x^2 \), allowing us to discuss function behavior with more complexity when parameters are involved.

Variation of the independent variable is key to understanding how a function behaves dynamically.

Square Root

The square root is an essential concept that frequently appears in mathematics. It refers to a value that, when multiplied by itself, gives the original number. In this exercise, the square root function is part of the expression \( 3 - \sqrt{y} \), which needs to be evaluated for the specified values of \( y \).

When dealing with square roots, it's important to remember:

• Square roots of perfect squares are whole numbers. For example, \( \sqrt{4} = 2 \) since \( 2 \times 2 = 4 \).
• Non-perfect squares result in irrational numbers. For example, \( \sqrt{0.25} = 0.5 \).

The square root function plays a significant role as it allows us to simplify expressions, which is often necessary for solving mathematical problems.

In the exercise, computing \( \sqrt{y} \) involves:

• Taking the square root of \( 4 \) resulting in \( 2 \).
• Taking the square root of \( 0.25 \) resulting in \( 0.5 \).
• Computing \( \sqrt{4x^2} \) to get \( 2x \), using the property \( \sqrt{a^2} = a \).

Mastering square roots is essential for manipulating and simplifying expressions involving radical terms.

Expression Simplification

Simplifying expressions is a crucial mathematical process that makes complex expressions more manageable by reducing them to their simplest form. In this exercise, after evaluating the function at each given substitution, expression simplification is performed to achieve the most reduced form of the function's value.

Expression simplification often follows these fundamental steps:

• Removing any unnecessary terms.
• Re-combining like terms.
• Performing arithmetic operations to reduce the expression thoroughly.

For instance, in this exercise:

• After substituting \( y = 4 \), the function becomes \( f(4) = 3 - 2 \), simplifying to \( 1 \).
• When \( y = 0.25 \), \( f(0.25) = 3 - 0.5 \) simplifies to \( 2.5 \).
• When \( y = 4x^2 \), \( f(4x^2) = 3 - 2x \), which is already in a simplified form.
  

By focusing on simplification, mathematicians can delve deeper into the problem-solving aspect, making the problem easier to handle and interpret. Always aim for the simplest form to understand and communicate solutions effectively.