Problem 37 - College Algebra and Calculus: An Applied Approach

Question

Evaluate the function at each specified value of the independent variable and simplify. \(f(y)=3-\sqrt{y}\) (a) \(f(4)\) (b) \(f(100)\) (c) \(f\left(4 x^{2}\right)\) (d) \(f(0.25)\)

Step-by-Step Solution

Answer

The evaluated function values are: \(f(4) = 1,\) \(f(100) = -7,\) \(f(4x^2) = 3 - 2x,\) and \(f(0.25) = 2.5.\)

Step 1: Evaluate function at \(f(4)\)

To evaluate the function at 'y=4', we substitute \(y=4\) into the function: \(f(4)=3-\sqrt{4}\). After simplifying, we find that: \(f(4) = 3 - \sqrt{4} = 3 - 2 = 1\).

Step 2: Evaluate function at \(f(100)\)

Now we substitute \(y=100\) into the function: \(f(100)=3-\sqrt{100}\). After simplifying, we find that: \(f(100) = 3 - \sqrt{100} = 3 - 10 = -7\).

Step 3: Evaluate function at \(f\left(4 x^{2}\right)\)

Now we substitute \(y=4x^2\) into the function: \(f(4x^2)=3-\sqrt{4x^2}\). After simplifying, we find that: \(f(4x^2) = 3 - \sqrt{4x^2} = 3 - 2x\).

Step 4: Evaluate function at \(f(0.25)\)

Now we substitute \(y=0.25\) into the function: \(f(0.25)=3-\sqrt{0.25}\). After simplifying, we find that: \(f(0.25) = 3 - \sqrt{0.25} = 3 - 0.5 = 2.5\).

Key Concepts

Simplifying Expressions

Simplifying expressions involves reducing an expression into its simplest form, making it easier to work with or understand. When simplifying, we focus on performing operations like addition, subtraction, multiplication, and division as far as possible, and rewriting complex expressions in a simpler equivalent form.
For the function given, simplification often involves reducing terms by calculating operations such as square roots or factoring variables.

• Example: In the function evaluation at \(f(4) = 3 - \sqrt{4}\), \(\sqrt{4}\) simplifies to 2. Hence, \(3 - \sqrt{4}\) becomes \(3 - 2\), which further simplifies to 1.
• Example: In the case \(f(100)\), \(\sqrt{100}\) simplifies to 10, transforming the expression to \(3 - 10\).

Consistently applying these operations helps reduce the expression to its simplest form, making it straightforward to assess or interpret.

Substitution

Substitution is a fundamental concept in math that involves replacing a variable in an expression with a given number or another expression. This allows for the numerical evaluation of functions or the transformation of algebraic expressions.
To evaluate a function, substitution is applied by simply inserting the given value into the place of the variable in an algebraic expression.

• Example: For the problem \(f(y) = 3 - \sqrt{y}\), to find \(f(4)\), replace \(y\) with 4 and simplify to get \(f(4) = 3 - 2 = 1\).
• Example: Similarly, when we compute \(f\left(4x^2\right)\), the substitution happens by replacing \(y\) with \(4x^2\), resulting in \(f(4x^2) = 3 - 2x\).

The process of substitution is crucial in math as it converts expressions into forms that can be solved or simplified.

Square Roots

Square roots are an essential part of algebra and involve finding a number which, when multiplied by itself, gives the original number. Square root operations can simplify expressions significantly by reducing a number into a more manageable form. In the given function \(f(y) = 3 - \sqrt{y}\), evaluating square roots is a key step in solving the problem:

• Example: When calculating \(f(0.25)\), the term \(\sqrt{0.25}\) becomes 0.5 because 0.5 multiplied by itself equals 0.25.
• Similarly, for \(y = 100\), \(\sqrt{100}\) results in 10, as 10 times 10 is 100.

Understanding how to calculate and simplify square roots plays a vital role in algebra and many other areas of mathematics, providing solutions a foundation to work on when solving equations or simplifying expressions.