Problem 59 - College Algebra with Modeling and Visualization

Question

Use f(x) to determine verbal, graphical and numerical representations. For the numerical representation use a table wish \(x=-2,-1,0,1,2\) Evaluate \(f(2).\) $$ f(x)=\sqrt{x+1} $$

Step-by-Step Solution

Answer

Verbal: \(\sqrt{x+1}\) defined for \(x\geq -1\). Graph: Starts at (-1,0). Numerically, \( f(2) = 1.73 \).

Step 1: Understanding the function

The function given is \( f(x) = \sqrt{x+1} \). This is a square root function which means it operates on the input \( x \) by adding 1 and then taking the square root of the result.

Step 2: Creating a Numerical Representation

To create a numerical representation, we will construct a table evaluating \( f(x) \) for \( x = -2, -1, 0, 1, 2 \).| \(x\) | \(f(x) = \sqrt{x+1}\) ||---|---|| -2 | \(\sqrt{-2+1} = \sqrt{-1}\), not real || -1 | \(\sqrt{-1+1} = \sqrt{0} = 0\) || 0 | \(\sqrt{0+1} = \sqrt{1} = 1\) || 1 | \(\sqrt{1+1} = \sqrt{2} \approx 1.41\) || 2 | \(\sqrt{2+1} = \sqrt{3} \approx 1.73\) |

Step 3: Verbal Interpretation

The function \( f(x) = \sqrt{x+1} \) takes a number \( x \) and computes the square root of the number resulting from \( x+1 \). The function is only defined for \( x \geq -1 \) because we cannot take the square root of a negative number in real numbers.

Step 4: Graphical Representation

To graph \( f(x) = \sqrt{x+1} \), plot the points found in the numerical table, starting from when \( x = -1 \) (where it equates to zero) and draw the curve that represents the function. The graph begins at the point (-1,0) and progressively increases, indicating a gradual rise along the \( x \)-axis.

Step 5: Evaluate \( f(2) \)

We already found \( f(2) \) from our table. Substituting \( x = 2 \) into the function yields \( f(2) = \sqrt{2+1} = \sqrt{3} \approx 1.73 \).

Key Concepts

Square Root Function

The square root function plays a crucial role in mathematics and has a distinctive form. In this case, the function given is \(f(x) = \sqrt{x+1}\). This function calculates the square root of the sum of \(x\) plus 1. The expression under the square root, \(x + 1\), is known as the radicand.
For any value of \(x\), you first add 1 to \(x\), then find the square root of this total. This function is only valid when \(x + 1 \geq 0\) because, in the realm of real numbers, you cannot take the square root of a negative number.
Therefore, the domain of \(f(x)\) is \(x \geq -1\). Hence, when exploring this function, ensure that you choose values of \(x\) that are greater than or equal to \(-1\).

Numerical Representation

To best understand a function like \(f(x) = \sqrt{x+1}\), a numerical representation is often helpful. Numerical representation involves calculating values at specific points. Let's compute \(f(x)\) at \(x = -2, -1, 0, 1, 2\) to form a table:

• For \(x = -2\), \(\sqrt{-2 + 1} = \sqrt{-1}\), which isn't real.
• For \(x = -1\), \(\sqrt{-1 + 1} = \sqrt{0} = 0\).
• For \(x = 0\), \(\sqrt{0 + 1} = \sqrt{1} = 1\).
• For \(x = 1\), \(\sqrt{1 + 1} = \sqrt{2} \approx 1.41\).
• For \(x = 2\), \(\sqrt{2 + 1} = \sqrt{3} \approx 1.73\).

Through this table, we begin to understand how \(f(x)\) behaves with different inputs, especially how it transitions from non-existence (at \(x = -2\)) to a gradual increase.

Graphical Representation

A graphical representation provides a visual insight into how a function behaves. For \(f(x) = \sqrt{x+1}\), the graph starts at the point \((-1, 0)\), as this is the first value of \(x\) where the function is defined. From here, as \(x\) increases, \(f(x)\) also increases.
Plotting the points from our numerical table onto a graph, you can see a curve that originates at \((-1,0)\) and gradually rises upward. This increasing trend reflects how the value of the square root grows larger as \(x\) becomes bigger.
The curve follows the pattern of other square root functions, showcasing a typical square root shape: starting slowly and increasing more sharply as you move along the \(x\)-axis.

Verbal Interpretation

Verbal interpretation translates the function \(f(x) = \sqrt{x+1}\) from mathematical notation to simple language. It tells us what the function does: For any chosen input \(x\), add 1 to it, and take its square root.
This interpretation helps identify the limitations and behavior of the function. Since you cannot have a square root of a negative number, \(x + 1\) must be non-negative, restricting \(x\) to values \(\geq -1\).
Verbally, we can say that from \(x = -1\) onward, the outputs represent the square root of each input \(x\) plus one. The further along the \(x\)-axis you go, the greater both the number you are adding to one and its root become.