Problem 43 - Precalculus with Limits

Question

In Exercises 37-52, 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) \(f(4x^2)\)

Step-by-Step Solution

Answer

Upon evaluating the functions we get \(f(4) = 1\); \(f(0.25) = 2.5\) and \(f(4x^2) = 3 - 2x\)

Step 1: Substitute y with 4

For the first part (a), substitute \(y\) with \(4\) in the function. The function then becomes: \(f(4) = 3 - \sqrt{4}\). Now, the square root of 4 is 2. So the expression becomes: \(f(4) = 3 - 2\).

Step 2: Substitute y with 0.25

For the second part (b), substitute \(y\) with \(0.25\) in the function. The function then becomes: \(f(0.25) = 3 - \sqrt{0.25}\). Now, the square root of 0.25 is 0.5. So the function simplifies to: \(f(0.25) = 3 - 0.5\).

Step 3: Substitute y with \(4x^2\)

For the third part (c), substitute \(y\) with \(4x^2\) in the function. The function then becomes: \(f(4x^2) = 3 - \sqrt{4x^2}\). Now, the square root of \(4x^2\) is \(2x\). So the function simplifies to: \(f(4x^2) = 3 - 2x\).

Key Concepts

Function Simplification

Understanding how to simplify functions is crucial for students when dealing with algebraic expressions. Function simplification involves reducing an expression to its most basic form without changing its value. In the context of the given exercise, simplifying a function means carrying out the operations within it and combining like terms if possible.

For example, when given a function like \(f(y) = 3 - \sqrt{y}\), simplifying it after substituting a given value involves two main steps. First, substitute the given value for the variable. Then, perform the operations, which in this case include subtracting the square root of the variable from a number. After these steps, you will achieve the simplified form of the function, which is easy to interpret and use for further calculations. Keeping the process orderly and step-by-step ensures accuracy and helps prevent common mistakes such as incorrect sign operations or mishandling of the square root.

Square Root Evaluation

Evaluating square roots is a fundamental skill in algebra that involves finding a number which, when multiplied by itself, gives the original number under the square root symbol. The square root of a number \(x\) is often denoted as \(\sqrt{x}\).

When encountering square roots in functions, it's important to remember that they represent the principal (non-negative) root. For instance, the square root of 4 is 2, and the square root of 0.25 is 0.5. In the exercise, evaluating the square root is key to simplifying the function and finding its value at a particular point. It's an example of how simple arithmetic operations can be used in function evaluation to move towards a simplified and understandable answer. Students should practice with different perfect and non-perfect squares to become proficient at quickly evaluating square roots.

Substituting Variables

Substituting variables in a function is a technique used to evaluate the function at a specific value. This involves replacing the variable in the function with a given number or another expression and performing the necessary arithmetic operations. It's a direct application of the concept of function, which is essentially a rule that assigns each input precisely one output.

In our exercise example, substitution plays a crucial role. Whether replacing the variable \(y\) with a number like 4 or with an expression like \(4x^2\), the goal is the same: to find out what the function equals for this particular input. Steps to do this effectively include writing the original function, replacing the variable with the given value, and then executing the necessary calculations. Attention to detail is important because a small mistake in substitution can lead to an incorrect result. Through frequent practice, students can master variable substitution and gain confidence in evaluating functions of various complexities.