Problem 37 - Algebra 2

Question

Graph. Find the domain and the range of each function. \(y=\sqrt{x}+7\)

Step-by-Step Solution

Answer

The domain of the function \(y=\sqrt{x}+7\) is [0, ∞), and the range is [7, ∞).

Step 1: Graph the function

First, plot the function \(y=\sqrt{x}+7\). Notice that the graph starts at (0,7) and extends infinitely to the right. Because you can't take the square root of a negative number, there are no values to the left of the y-axis.

Step 2: Determine the Domain

Looking at the graph and considering the type of function, we see that x (the input) can be any value that is 0 or greater, but not less than 0. Therefore, the domain is \(x \geq 0\) or in interval notation, [0, ∞).

Step 3: Determine the Range

For the range of the function, which is the set of output values (y-values), we see from the graph that y can take any value that is 7 or greater. Therefore, the range is \(y \geq 7\) or in interval notation, [7, ∞).

Key Concepts

Domain of a Function

In mathematics, a function's domain is the complete set of possible input values, or "x-values," that the function can accept. Let's take the example from the given exercise, the function is defined as \( y = \sqrt{x} + 7 \). For this function, we need to identify for which values of \(x\) this function is valid, meaning where the expression inside the square root remains non-negative.

• The expression under the square root, \( x \), must be 0 or greater because the square root of a negative number is not a real number.
• Thus, the smallest value \( x \) can be is 0.
• This means that the domain for this particular function is all real numbers from 0 to infinity.

In interval notation, we express this conclusion as \([0, \infty)\). Always remember, when determining a function's domain, think about the values of \( x \) that make the function undefined or not real. For square root functions, this means looking for conditions where the inside of the square root is defined.

Range of a Function

The range of a function is essentially the set of all possible output values, or "y-values," that you get when you plug every value in the domain into the function. For the function \( y = \sqrt{x} + 7 \), we need to determine the smallest and largest output values when \( x \) spans all values in the domain.

• Since we know that the lowest value of \( x \) is 0, plug it into the function: \( y = \sqrt{0} + 7 = 7 \).
• This calculation shows that the smallest value \( y \) can be is 7.
• As \( x \) increases, \( \sqrt{x} \) also increases, resulting in bigger \( y \)-values.

Since \( \sqrt{x} \) gets larger with increasing \( x \), \( y = \sqrt{x} + 7 \) will also continue to increase without bound. Thus, there is no upper limit to the values it can output. Therefore, the range is expressed in interval notation as \([7, \infty)\). Understanding the range involves visualizing the function or referring back to the graphed data to observe how high or low the function's output can go.

Square Root Function

A square root function is a type of function that involves the square root of a quantity, usually of the form \( y = \sqrt{x} \) or, in our example, \( y = \sqrt{x} + c \), where \( c \) is a constant. Let's break down its properties and behavior using the function \( y = \sqrt{x} + 7 \) from our exercise.

• The graph of a basic square root function \( y = \sqrt{x} \) starts at the origin (0,0) and exhibits a gentle increase, forming a curve that extends infinitely to the right but not to the left, as it's undefined for \( x < 0 \).
• In our example, since there is an addition of 7 to \( \sqrt{x} \), the whole graph shifts upwards by 7 units.
• This gives the graph a new starting point at (0,7), manifesting the domain \( x \geq 0 \) and affecting the range to be \( y \geq 7 \).

Square root functions are particularly easy to track visually due to their distinctive rightward, upward-curving pattern. Understanding how transformations like these affect function graphs is crucial to mastering how different function types behave and connect algebraic rules with their graphical representations.