Problem 49 Graph. Find the domain and the r... [FREE SOLUTION]

Chapter 7: Problem 49

Graph. Find the domain and the range of each function. \(y=-1-\sqrt{4 x+20}\)

Short Answer

Expert verified

The domain of the function \(y=-1-\sqrt{4 x+20}\) is \([-5, +\infty)\), and the range is \(-\infty, -1]\).

Step by step solution

Find the domain of the function

To find the domain, we have to solve the inequality \(4 x+20 \geq 0\), because the quantity under the square root should not be negative. We solve for \(x\) to get: \[x \geq -5\] So the domain is \([-5, +\infty)\].

Find the range of the function

In general, the square root function \(y=\sqrt{x}\) has a range of \([0, +\infty)\) but in the given function, the square root is negated and then subtracted by 1. The negation flips it over the x-axis, and the subtraction shifts it down by 1 unit. Therefore, the range of the function is \(-\infty, -1]\).

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Key Concepts

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

Graphing Functions

Graphing functions is the process of visually representing algebraic equations on a coordinate plane. This helps us understand how a function behaves, including its domain, range, and any transformations it may have undergone.
When graphing, we usually start with a known basic function, such as a line or a parabola, and then apply any transformations.
There are several key elements to consider when graphing a function:

**Domain:** This defines all the possible values of the input (usually x) for which the function is defined. For example, if the function involves a square root, the domain will only include values that do not make the expression under the square root negative.
**Range:** This is the set of possible output values (usually y) that the function can produce. The transformations applied to the function affect its range.
**Intercepts:** These are points where the function crosses the x or y-axes. Knowing these can help to plot the function initially.

Graphing allows you to visually analyze the transformation effects and identify properties of the function, such as whether it is increasing, decreasing, or has any symmetries.

Square Root Functions

Square root functions are a type of radical function that involves the square root of a variable. These functions are generally expressed in the form of \( y = \sqrt{x} \).
The basic square root function has several important characteristics:

**Domain:** The expression inside the square root must be non-negative; hence, for the basic form \( y = \sqrt{x} \), the domain is \([0, +\infty)\).
**Range:** The output, or y-values, are also non-negative, providing a range of \([0, +\infty)\).
**Graph Shape:** The graph of a square root function is a curve that starts at the origin \((0,0)\) and rises to the right, becoming less steep.

In the equation \( y = -1 - \sqrt{4x + 20} \), the expression under the square root \(4x + 20\) must be non-negative, which influences the domain. Additionally, the negative sign before the square root indicates a reflection of the curve over the x-axis, and the subtraction of 1 shifts the graph downward.

Function Transformation

Function transformation involves altering the basic form of a function to shift, reflect, stretch, or compress it. This can significantly change the way a function looks on a graph.
The main types of transformations are:

**Vertical Shifts:** Adding or subtracting a constant from the y-value moves the graph up or down. For example, \( y = f(x) + c \) shifts the function upwards by \(c\) units.
**Horizontal Shifts:** Adding or subtracting a constant inside the function argument moves the graph sideways. \( y = f(x + c) \) shifts the function to the left by \(c\) units.
**Reflections:** Multiplying a function by a negative value flips it over an axis. For example, \( y = -f(x) \) reflects it over the x-axis.
**Vertical Stretch/Compression:** Multiplying the function by a factor greater or less than 1 stretches or compresses it vertically.
**Horizontal Stretch/Compression:** Similar concepts apply horizontally.

In the problem \( y = -1 - \sqrt{4x + 20} \), there is a vertical reflection due to the negative sign in front of the square root and a downward shift by 1 due to the subtraction. This transformation affects the range, leading to a new set of y-values.

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91影视

Graph. Find the domain and the range of each function. \(y=-1-\sqrt{4 x+20}\)

Short Answer

Step by step solution

Find the domain of the function

Find the range of the function

Key Concepts

Graphing Functions

Square Root Functions

Function Transformation

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