Question

\(f(x)=\sqrt{x+3}\)

Short answer

The domain of the function \(f(x)=\sqrt{x+3}\) is \([-3, \infty)\). The value of \(f(x)\) for any given \(x\) can be found by plugging \(x\) into the function. For example, \(f(4) = \sqrt{7}\). The graph of the function starts from the point (-3, 0) and continues upwards and towards the right.

Step-by-step solution

Determine the Domain

The domain of a function is the set of all permissible input values (x-values). For this function, we need to consider that the square root of a negative number is undefined. Therefore, the inside of the square root, \(x + 3\), must be greater than or equal to 0. Solving this inequality, we find: \(x + 3 >= 0\), thus \(x >= -3\). So the domain of the function is \([-3, \infty)\).

Find the Value of Function for a Given x-value

To find the value of function for any permissible \(x\) value, we substitute \(x\) into the function and solve. For example, to find \(f(4)\), we substitute \(x = 4\) into the function: \(f(4) = \sqrt{4 + 3} = \sqrt{7}\).

Graph the function

The graph of the function \(f(x)=\sqrt{x+3}\) is a square root function which is shifted three units to the left. It starts from the point (-3, 0) and continues upwards and towards the right.

Key concepts

Square Root Function

The square root function is a type of radical function, specifically the one where you are finding the principal square root of a number. In its simplest form, it's represented as \( f(x) = \sqrt{x} \). However, the square root function can be transformed by altering its formula; for example, \( f(x) = \sqrt{x + 3} \) shifts the standard square root graph horizontally to the left by 3 units.

This transformation impacts the domain of our function. The domain is the set of all possible input values (x-values) for which the function is defined. As you can't take the square root of a negative number without delving into complex numbers (which are not within the scope of this function), the function \( f(x) = \sqrt{x + 3} \) requires that \( x + 3 \geq 0 \). Solving this inequality gives us the domain \( x \geq -3 \), which is the set of all real numbers from -3 to positive infinity, written in interval notation as \( [-3, \infty) \).

The square root function graph is shaped like a sideways 'L' on its back, reflective of its increasing nature; when shifted, it retains its basic shape but starts or turns from a different point on the graph.

Inequality Solving

When solving inequalities, our goal is to find the range of values that satisfy the condition of being either less than, greater than, less than or equal to, or greater than or equal to a certain number or expression. It's similar to solving an equation, but instead of finding a single solution, we find a range of solutions.

Consider the inequality \( x + 3 \geq 0 \). Here, we want to find all the values of x that make the expression true. The first step is to isolate x by subtracting 3 from both sides to yield \( x \geq -3 \). The solution to this inequality is not just a single number, but every number greater than or equal to -3. When available, representing these solutions on a number line helps visualize the range of acceptable values.

In the context of the square root function mentioned previously, solving the inequality is crucial for determining the domain. Without understanding the inequality, we can't correctly graph the function or evaluate it for given x-values.

Graphing Functions

Graphing functions provides a visual representation of the relationship between the independent variable (usually x) and the dependent variable (usually y). For the square root function \( f(x) = \sqrt{x + 3} \), we graph it by plotting points that satisfy the function and then connecting these points to illustrate the continuous nature of the graph.

To begin graphing, we first identify the starting point. In this case, because the domain of our function begins at \( x = -3 \), we start our graph at the point (-3, 0). From there, as the value of x increases, the value of the function also increases, albeit at a decreasing rate. As a result, the graph gradually levels off as x becomes much larger.

The graph of the square root function is unique in that it represents a half-parabola lying on its side, opening to the right. Remember that the function will include all points where \( x \) is greater than or equal to -3, and the corresponding y-values are those obtained by taking the square root of \( x + 3 \). By accurately plotting a sufficient number of points and connecting them smoothly, the shape of the graph will emerge, illustrating the functional relationship between x and y.