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Chapter 1: Problem 37

\(33-44\) Find the domain and sketch the graph of the function. $$g(x)=\sqrt{x-5}$$

Short Answer

Expert verified
The domain is \([5, \infty)\). Graph: Starts at (5, 0), curves upwards.

Step by step solution

01

Understand the Function

The function given is a square root function: \( g(x) = \sqrt{x-5} \). Square root functions are defined only when the expression under the square root is non-negative.
02

Set up the Domain Condition

To find the domain, set the expression inside the square root to be greater than or equal to zero: \( x-5 \geq 0 \).
03

Solve for x

Solve the inequality obtained in Step 2. So, \( x \geq 5 \).
04

Write the Domain

The domain of \( g(x) \) is all values of \( x \) such that \( x \geq 5 \). In interval notation, this is \([5, \infty)\).
05

Sketch the Graph

Plot the basic shape of \( g(x) = \sqrt{x-5} \), which starts at the point (5, 0) and moves upwards to the right along a curve, since square roots graph as a curve extending to infinity.

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Key Concepts

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

Understanding Square Root Functions
In the realm of mathematics, a square root function is a fascinating concept. It is expressed as \( g(x) = \sqrt{x-c} \), where \( c \) is a constant that shifts the graph horizontally. The defining characteristic of a square root function is that it only exists for values where the expression under the square root is non-negative. This is because the square root of a negative number is not defined within the set of real numbers.
For our function, \( g(x) = \sqrt{x-5} \), we are interested in the values of \( x \) that ensure \( x-5 \geq 0 \). Therefore, square root functions often begin at a certain point on the x-axis and curve upwards to the right, creating a distinct shape.
Solving Inequalities to Find the Domain
Inequality solving is a crucial process in determining the domain of functions like square roots. The domain of a function is the set of all possible inputs (x-values) that will produce a real number output. To find where \( g(x) = \sqrt{x-5} \) is defined, we set up the inequality from the expression inside the square root: \( x - 5 \geq 0 \). This inequality must hold true for us to find valid x-values.
Solving this inequality, we add 5 to both sides, resulting in \( x \geq 5 \). This tells us that only those x-values which are 5 and above will work in this function, ensuring the expression under the square root is non-negative.
Expressing the Domain in Interval Notation
Once the inequality \( x \geq 5 \) is solved, we can express the domain of the function using interval notation. Interval notation is a shorthand way to write the set of numbers that satisfy an inequality.
For our problem, \( x \geq 5 \) means x starts at 5 and goes towards infinity. This is represented in interval notation as \([5, \infty)\). The bracket \([ \) at 5 means that 5 is included in the domain, while the parenthesis \( ) \) at infinity suggests it extends indefinitely.
This concise form of notation helps in easily understanding and communicating the range of x-values for which the function is valid.
Graph Sketching for Visual Understanding
Sketching the graph of a function is an excellent way to visualize and understand its behavior. For the function \( g(x) = \sqrt{x-5} \), we start our graph at the point (5, 0). This is because when \( x = 5 \), the expression inside the square root turns zero, giving \( g(x) = 0 \).
As x increases beyond 5, the value of \( g(x) \) will also increase, following a gentle curve upwards to the right. The graph of a square root function typically reflects a smooth, continuous increase and forms what looks like half of a sideways parabola.
  • The curve will never dip below the x-axis since square roots of non-negative numbers yield non-negative results.
  • The graph progresses infinitely to the right, indicating that \( x \) can grow without bound.
This sketch reinforces our understanding of the domain and range, visually showing that \( g(x) \) is defined for all \( x \geq 5 \).

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Most popular questions from this chapter

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