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Chapter 0: Problem 20

\(y=\sqrt{x+5}\)

Short Answer

Expert verified
The short answer is that the square root function \(y=\sqrt{x+5}\) starts at point (-5,0) and increases as \(x\) increases. The important aspect of this function to note is that it does not exist for \(x < -5\). Because you cannot take the square root of a negative number in the real number set, thus it does not exist for \(x < -5\).

Step by step solution

01

Identify the function type

The given function \(y=\sqrt{x+5}\) is a square root function. It is in the form \(y= \sqrt{x+c}\) where \(c\) is a constant. At this stage, the function's basic characteristic and graph shape should be recognized.
02

Compute function values for some \(x\) values

To understand the function, calculate a few values. For example, for \(x=-5\), using the function, \(y=\sqrt{-5+5}=0\). Thus, when \(x=-5\), \(y=0\). One could try out more \(x\) values such as \(x=0\) or \(x=5\) to understand its variation.
03

Graph the function

Plot the computed points on a graph. The square root function graph will begin from the point (-5,0) (as calculated) and will move upward and to the right. The plot will give a fair idea of the function's behaviour.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Function Graphing
Graphing a function helps us visualize its behavior. For the function \(y = \sqrt{x+5}\), we observe a key feature:
  • The square root function starts at a specific point and moves infinitely in a particular direction.
  • The graph of \(y = \sqrt{x+5}\) starts at the point \((-5, 0)\).
  • This is because the expression inside the square root becomes zero at \(x = -5\).
In graphing this function, plot several points to capture the path of the curve accurately.
Starting with the known point \((-5, 0)\), consider additional points like \(x = 0\) corresponding to \(y = \sqrt{5} = \sqrt{5}\).
Notice how each increase in \(x\) results in a smooth upward curve, forming a half-parabola opening to the right. This indicates the function is defined for \(x \geq -5\), and helps convey the domain of this square root function.
Evaluating Functions
Evaluating a function involves finding output values for specific input values. With \(y = \sqrt{x+5}\), this means substituting various \(x\) values into the function to see what \(y\) comes out.
  • If \(x = -5\), \(y\) equals zero since \(y = \sqrt{-5+5} = \sqrt{0}\).
  • Trying \(x = 0\), we get \(y = \sqrt{0+5} = \sqrt{5}\), which is a little over two.
  • For \(x = 5\), \(y = \sqrt{10}\).
When evaluating, make sure \(x + 5 \geq 0\) for real number results, meaning \(x\) must be greater than or equal to -5.
This gives the domain of our function as \(x \geq -5\). Evaluating functions allows us to understand not just isolated points but the overall behavior and restrictions of the function.
Algebraic Functions
Algebraic functions are built from basic operations like addition, subtraction, multiplication, division, and taking roots. The square root function \( y = \sqrt{x+5} \) is a prime example because it involves the operation of finding a square root.

Such functions can take many forms but they often share certain traits:
  • They arise from basic arithmetic operations performed on variables.
  • In \(y = \sqrt{x+5}\), addition and taking the square root are the elements of the function.
  • These functions can have different domains depending on the operation inside. For square roots, the inside expression must be non-negative.
Algebraic functions like this are a foundation in mathematics, helping us describe relationships and changes between quantities.
Understanding these fundamental elements makes other complex concepts more approachable. Recognizing these underlying algebraic operations provides insights that make more advanced topics easier to grasp.

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Most popular questions from this chapter

True or False? In Exercises 85 and 86, determine whether the statement is true or false. Justify your answer. Proof Prove that if \(f\) is a one-to-one odd function, then \(f^{-1}\) is an odd function.

In Exercises 55-68, determine whether the function has an inverse function. If it does, find the inverse function. $$ f(x)=|x-2|, \quad x \leq 2 $$

Find the average rate of change of the function from \(x_{1}\) to \(x_{2}\). $$ \begin{array}{cc} \text { Function } & x \text {-Values } \\ \(f(x)=-\sqrt{x-2}+5 &\quad x_{1}=3, x_{2}=11\) \end{array} $$

In Exercises 75-78, use the functions given by \(f(x)=x+4\) and \(g(x)=2 x-5\) to find the specified function. $$ (g \circ f)^{-1} $$

In Exercises 55-68, determine whether the function has an inverse function. If it does, find the inverse function. $$ p(x)=-4 $$
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