Problem 44 - Precalculus with Limits

Question

In Exercises 37-52, evaluate the function at each specified value of the independent variable and simplify. \(f(x) = \sqrt{x+8}+2\) (a) \(f(-8)\) (b) \(f(1)\) (c) \(f(x-8)\)

Step-by-Step Solution

Answer

(a) \(f(-8) = 2\), (b) \(f(1) = 5\), (c) \(f(x-8) = \sqrt{x} + 2\)

Step 1: Evaluate \(f(-8)\)

Replacing \(x\) with \(-8\) in the original function gives \(f(-8) = \sqrt{-8+8}+2 = \sqrt{0}+2 = 2\).

Step 2: Evaluate \(f(1)\)

Replacing \(x\) with \(1\) in the original function gives \(f(1) = \sqrt{1+8}+2 = \sqrt{9}+2 = 3+2 = 5\).

Step 3: Evaluate \(f(x-8)\)

Replacing \(x\) with \(x-8\) in the original function gives \(f(x-8) = \sqrt{(x-8)+8}+2 = \sqrt{x} + 2\).

Key Concepts

Square Root Function

A square root function is a type of function that involves the square root of an expression. It is commonly written in the form \(f(x) = \sqrt{x}\). In our exercise, the function given is \(f(x) = \sqrt{x+8} + 2\). This means that for any value of \(x\), you take the square root of \(x+8\) and then add 2. The square root function is characterized by its radical, which makes it important to focus on understanding its properties:

• The square root function is only defined for non-negative values within the radical, which means \(x + 8 \geq 0\). This ensures that the result is always a real number.
• As \(x\) increases, the result of the square root typically increases at a decreasing rate. This gives the function a gentle upward slope.
• The standard square root function, \(\sqrt{x}\), starts at the origin \((0,0)\) and curves upwards to the right, but in our case, we have an additional constant (+2) which shifts the function vertically upwards.

Independent Variable

The independent variable in a function is the variable that you can change freely. It is often denoted as \(x\). In any mathematical relationship, the value of the dependent variable, which is controlled by an equation, depends on the variable chosen independently. In our exercise \(f(x) = \sqrt{x+8} + 2\), \(x\) represents the independent variable. Here’s why it's crucial:

• The independent variable determines the input of our function. For example, in \(f(-8)\), \(-8\) is the input for \(x\).
• The choice of \(x\) influences the output of the function, or \(f(x)\), which is essentially why we evaluate functions at particular values of \(x\).
• Recognizing the independent variable helps us understand the flexibility we have in changing it to see how the function behavior evolves accordingly.

Function Simplification

Function simplification is the process of making a function easier to evaluate or understand. It involves rewriting the function in its simplest form without changing its underlying meaning. This process was used in each step of evaluating the given function.When you evaluated \(f(x) = \sqrt{x+8}+2\) at specific values of \(x\), here's how simplification took place:

• For \(f(-8)\), substituting \(x = -8\) results in \(\sqrt{0} + 2\) which simplifies directly to 2. This step involves simple arithmetic: \(\sqrt{0} = 0\).
• For \(f(1)\), substituting \(x = 1\) gives \(\sqrt{9} + 2\) which simplifies to 5 since \(\sqrt{9} = 3\).
• In \(f(x-8)\), substituting \(x = x-8\) simplifies to \(\sqrt{x} + 2\) by canceling out the \(+8\) and \(-8\) terms under the radical.

Function simplification often helps in better understanding the behavior of the function and in this context, it makes evaluating at various \(x\)-values straightforward and clear.