Problem 71 - College Algebra

Question

For the following exercises, evaluate the function \(f\) at the values \(f(-2), f(-1), f(0), f(1),\) and \(f(2)\). $$ f(x)=3+\sqrt{x+3} $$

Step-by-Step Solution

Answer

The evaluated values are \(f(-2)=4, f(-1)\approx 4.414, f(0)\approx 4.732, f(1)=5, f(2)\approx 5.236\).

Step 1: Evaluate f(-2)

To evaluate the function at \(x = -2\), substitute -2 into the function: \[f(-2) = 3 + \sqrt{-2 + 3}\]Simplify inside the square root:\[f(-2) = 3 + \sqrt{1}\]Calculate the square root and add:\[f(-2) = 3 + 1 = 4\]

Step 2: Evaluate f(-1)

To find \(f(-1)\), substitute -1 into the function: \[f(-1) = 3 + \sqrt{-1 + 3}\]Simplify inside the square root:\[f(-1) = 3 + \sqrt{2}\]Since \(\sqrt{2}\) is approximately 1.414, the expression becomes:\[f(-1) \approx 3 + 1.414 = 4.414\]

Step 3: Evaluate f(0)

To find \(f(0)\), substitute 0 into the function:\[f(0) = 3 + \sqrt{0 + 3}\]Simplify inside the square root:\[f(0) = 3 + \sqrt{3}\]Since \(\sqrt{3}\) is approximately 1.732, the expression becomes:\[f(0) \approx 3 + 1.732 = 4.732\]

Step 4: Evaluate f(1)

To find \(f(1)\), substitute 1 into the function:\[f(1) = 3 + \sqrt{1 + 3}\]Simplify inside the square root:\[f(1) = 3 + \sqrt{4}\]Calculate the square root and add:\[f(1) = 3 + 2 = 5\]

Step 5: Evaluate f(2)

To find \(f(2)\), substitute 2 into the function:\[f(2) = 3 + \sqrt{2 + 3}\]Simplify inside the square root:\[f(2) = 3 + \sqrt{5}\]Since \(\sqrt{5}\) is approximately 2.236, the expression becomes:\[f(2) \approx 3 + 2.236 = 5.236\]

Key Concepts

Algebraic Functions

When dealing with algebraic functions like the one given in the exercise, you're essentially working with expressions that are constructed using constants, variables, and a limited set of operations: addition, subtraction, multiplication, division, and extraction of roots. In our exercise, the function is defined as follows:\[f(x) = 3 + \sqrt{x+3}\]This means that for any given value of \(x\), you need to substitute that value into the function. Algebraic functions can be used to model a wide range of real-world situations, which is why understanding them is so important. Keep a few tips in mind:

• Recognize the basic structure of the function, which in this case includes a square root.
• Take note that these functions can only be evaluated for real numbers that keep the square root sign valid (non-negative).
• The function's behavior might change with different values of \(x\), reflecting in the shape or position of its graph.

Grasping how algebraic functions work helps us translate complex problems into manageable, solvable equations.

Square Roots

Square roots, symbolized by \( \sqrt{} \), are a way of expressing a number that, when multiplied by itself, gives the original number inside the root. The function in our exercise incorporates a square root as part of its makeup:- For example, if you look at the square root within \(f(x) = 3 + \sqrt{x+3}\), you're specifically targeting the expression inside the root.When calculating square roots:

• The contents under the square root must be non-negative for real number outcomes.
• Recognize common roots, such as \(\sqrt{1} = 1\) and \(\sqrt{4} = 2\), as they make calculations easier.
• Approximations like \(\sqrt{2} \approx 1.414\) or \(\sqrt{3} \approx 1.732\) may be used for more complex roots.

Working successfully with square roots requires some familiarity with both perfect and approximate root calculations, easing the function evaluation process.

Substitution Method

The substitution method is a fundamental technique in evaluating functions, especially useful in exercises like the one presented. Here's how you apply it in a practical setting:- Begin with the given function, such as \(f(x) = 3 + \sqrt{x+3}\).For any specified value of \(x\):

• Substitute the value directly into the function wherever \(x\) appears.
• Simplify the resulting expression by handling operations within any parentheses first, move to apply the square root, and finally perform any addition or other operations left.
• Use rough approximations for square roots when necessary for tidy linear expressions.

In practice, substituting helps you determine function values with precision, aiding in graphing, analyzing or applying mathematical models to real-world scenarios. It is a simple yet powerful tool for determining output values for a range of inputs seamlessly.