Question

Find the domain and the range of the function. Then sketch the graph of the function. $$y=2 \sqrt{4 x+10}$$

Short answer

The domain of the function \(y = 2 \sqrt{4x + 10}\) is [-2.5, \(\infty\)] and its range is [0, \(\infty\)]. The graph starts at point (-2.5, 0) and curves upwards, getting steeper as \(x\) increases.

Step-by-step solution

Identify the domain

The domain of a function is all the possible values of \(x\) for which the function is defined. A square root function is only defined when the input is greater or equal to zero. So, we need to solve the inequality \(4x + 10 \geq 0\). Solving this inequality gives \(x \geq -\frac{10}{4} = -2.5 \). Therefore, the domain of the function is [-2.5, \(\infty\)].

Determine the range

The range of a function is the set of all possible output values (\(y\)). As the square root function only returns non-negative values, the minimum of \(y\) will be when \(x\) equals -2.5 (as per our domain). Substituting -2.5 into the function, we get \(y = 2 \sqrt{4(-2.5) + 10} = 0\). Since the square root function increases as \(x\) increases, the maximum of \(y\) is \(\infty\). Therefore, the range of the function is [0, \(\infty\)].

Sketching the graph

Now we know the domain and the range, we can sketch the graph. Ignoring the multiplier 2 for a moment, the graph of the square root starts at the point (-2.5, 0) and increases slowly at first and then more rapidly as \(x\) gets larger. The multiplier two 'stretches' the graph vertically by a factor of two. This means for each \(x\), the value of \(y\) will be twice as large as it would be for the standard square root function.

Key concepts

Square Root Function

The square root function, often represented as \( f(x) = \sqrt{x} \), plays a pivotal role in understanding the behavior of more complex functions involving square roots. When dealing with these functions, it's essential to establish their domain—the set of all possible input values, \( x \), for which the function is defined. For a square root function, the domain is limited to non-negative numbers since the square root of a negative number is not a real number.

For the given function \( y = 2 \sqrt{4x + 10} \), we have to find the smallest value of \( x \) such that \( 4x + 10 \) is non-negative. This leads to an inequality \( 4x + 10 \geq 0 \) to find the lower boundary of the domain. In this scenario, the domain starts at \( x = -2.5 \) and extends to infinity, as there's no upper bound to the values \( x \) can take. Additionally, the multiplier '2' in front of the square root doesn't affect the domain but does influence the range and shape of the graph.

Inequalities

Inequalities are mathematical expressions involving the symbols \( >, <, \geq, \leq \) that show the relationship between two values. They play a crucial role in determining the domain and range of functions. When an inequality is presented in a function, such as \( 4x + 10 \geq 0 \) in our square root function, we are essentially looking for all the possible values of \( x \) that make the inequality true.

To solve the inequality, we isolate the variable on one side, resulting in \( x \geq -2.5 \). The solution to an inequality is often shown on a number line or with interval notation, which, in our case, would be \( [-2.5, \infty) \). Inequalities not only determine which values are in the domain but also shape the graph of the function, as only those x-values that satisfy the inequality will be included in the graph.

Graphing Functions

Graphing is a powerful tool for visualizing the behavior of mathematical functions. It translates the abstract concepts of domain and range into a concrete visual representation. When graphing the square root function \( y = 2 \sqrt{4x + 10} \), we start with identifying the domain and range. The graph begins at the vertex, which corresponds to the smallest \( x \) in the domain, and increases from there—this is the point \( (-2.5, 0) \) in our exercise.

As we plug in increasing \( x \) values, the corresponding \( y \) values are calculated using the function's formula. Since the function involves a square root, the \( y \) values will increase at a decreasing rate. Multiplying the output of the square root by 2 'stretches' the function vertically, which means that the graph will be taller and steeper than the basic square root graph. The result is a graph that starts from the vertex and curves upward as \( x \) moves towards infinity, marking a clear visualization of the function's increasing behavior.