Problem 71 - College Algebra with Corequisite Support

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

f(-2) = 4, f(-1) ≈ 4.41, f(0) ≈ 4.73, f(1) = 5, f(2) ≈ 5.24.

Step 1: Evaluate f(-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:\[ f(-2) = 3 + 1 = 4 \]

Step 2: Evaluate f(-1)

Substitute -1 into the function: \[ f(-1) = 3 + \sqrt{-1 + 3} \]Simplify inside the square root:\[ f(-1) = 3 + \sqrt{2} \]Since \( \sqrt{2} \approx 1.41 \), approximate the value:\[ f(-1) \approx 3 + 1.41 = 4.41 \]

Step 3: Evaluate f(0)

Substitute 0 into the function: \[ f(0) = 3 + \sqrt{0 + 3} \]Simplify inside the square root:\[ f(0) = 3 + \sqrt{3} \]Since \( \sqrt{3} \approx 1.73 \), approximate the value:\[ f(0) \approx 3 + 1.73 = 4.73 \]

Step 4: Evaluate 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:\[ f(1) = 3 + 2 = 5 \]

Step 5: Evaluate f(2)

Substitute 2 into the function: \[ f(2) = 3 + \sqrt{2 + 3} \]Simplify inside the square root:\[ f(2) = 3 + \sqrt{5} \]Since \( \sqrt{5} \approx 2.24 \), approximate the value:\[ f(2) \approx 3 + 2.24 = 5.24 \]

Key Concepts

Square Root

The square root is a mathematical operation that finds a number which, when multiplied by itself, gives the original number. For instance, the square root of 9 is 3, since 3 x 3 = 9. In math notation, it is represented by the radical symbol (\(\sqrt{}\)).
In our function evaluation example, we frequently encounter square roots, such as \(\sqrt{1}\), \(\sqrt{2}\), \(\sqrt{3}\), \(\sqrt{4}\), and \(\sqrt{5}\).

• Understanding square roots is crucial because it helps in evaluating expressions and functions where the variable inside the square root changes.
• In practical terms, finding the square root of a number involves determining what number squared results in the original number.

To simplify calculations or mathematical expressions involving square roots, ensure that the number is a perfect square or approximate the square root otherwise.

Substitution

Substitution is the process of replacing a variable with a specific value. It is commonly used in mathematics when evaluating expressions or functions.
In our exercise, we substitute various values for \(x\) into the function \( f(x) = 3 + \sqrt{x+3} \).

• For example, when substituting \(-2\) in place of \(x\), the expression becomes \( f(-2) = 3 + \sqrt{-2 + 3} \).
• This technique allows us to find the result or output of the function for different inputs or values of \(x\).

Substitution is especially useful in situations where a function or equation needs to be evaluated at specific points. By substituting specific values, one can determine the output at those given points.

Approximation

Approximation involves finding a value that is close enough to the exact solution, especially when exact solutions are difficult to determine. It's crucial in mathematics when dealing with irrational numbers like square roots that are not perfect squares.
In our function, expressions such as \(\sqrt{2}\), \(\sqrt{3}\), and \(\sqrt{5}\) give irrational numbers.

• The value \(\sqrt{2} \approx 1.41\) doesn't precisely equal \(\sqrt{2}\), but it is a close estimate.
• This helps when we need a practical, usable number for further calculations or insights.

Approximation is helpful because it provides a manageable number to work with, allowing us to continue solving problems without requiring highly complex or exact numbers at every step.

Simplification

Simplification in mathematics involves making an expression easier to understand or solve. This often means reducing an expression to its simplest form.
During our function evaluations, we simplify expressions inside a square root before further calculations. For example, \(\sqrt{-2+3}\) simplifies to \(\sqrt{1}\).

• By simplifying the expression within the square root, we make it more straightforward to compute subsequent operations.
• The process often involves combining like terms, factoring, or canceling common factors.

Simplification is essential because it clarifies the problem, making it easier to solve and understand. It removes complex or extraneous information to reveal the core components of an equation or expression.