Chapter 12: Q 25. (page 916)

In Exercise, evaluate the given function at the specified points in the domain, and then find the domain and range of the function.
$g (x, y) = \frac{\ln (x y)}{\sqrt{1 - x}}, (- e, - 1), (\frac{1}{2}, 2)$

Short Answer

Expert verified

Domain is ${(x, y) ∣ x < 0; y < 0} \cup {(x, y) ∣ 0 < x < 1; y > 0}$

The range is R.

localid="1653905469596" $\begin{array}{rcl} g (- e, - 1) & = & \frac{1}{\sqrt{1 + e}} \\ g (- e, - 1) & = & 0 \end{array}$

Step by step solution

Step 1. Given information

Function is $g (x, y) = \frac{\ln (x y)}{\sqrt{1 - x}}$

Step 2. Explanation

$\begin{array}{rcl} g (- e, - 1) & = & \frac{\ln ((- e) (- 1))}{\sqrt{1 - (- e)}} \\ = & \frac{\ln (e)}{\sqrt{1 + e}} \\ = & \frac{1}{\sqrt{1 + e}} \\ g (- e, - 1) & = & \frac{\ln ((\frac{1}{2}) (2))}{\sqrt{1 - (\frac{1}{2})}} \\ = & \frac{\ln (1)}{\sqrt{\frac{1}{2}}} \\ = & 0 \end{array}$

Another goal is to figure out the function's domain and range.

A logarithmic function is used in the function. The positive real numbers are the domain of a logarithmic function. Make the logarithmic function's parameter greater than 0.

$x y > 0$

$x, y > 0$ or $x, y < 0$

$\begin{array}{rcl} 1 - x & > & 0 \\ x & < & 1 \end{array}$

Domain is ${(x, y) ∣ x < 0; y < 0} \cup {(x, y) ∣ 0 < x < 1; y > 0}$

Range is R as there is no constraint over the output values.

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In Exercise, evaluate the given function at the specified points in the domain, and then find the domain and range of the function.
$g (x, y) = \frac{\ln (x y)}{\sqrt{1 - x}}, (- e, - 1), (\frac{1}{2}, 2)$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Explanation

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In Exercise, evaluate the given function at the specified points in the domain, and then find the domain and range of the function.gx,y=lnxy1-x,-e,-1,12,2

Short Answer

Step by step solution

Step 1. Given information

Step 2. Explanation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Calculus

Geometry

Discrete Mathematics

Mechanics Maths

Logic and Functions

Study anywhere. Anytime. Across all devices.

In Exercise, evaluate the given function at the specified points in the domain, and then find the domain and range of the function.
$g (x, y) = \frac{\ln (x y)}{\sqrt{1 - x}}, (- e, - 1), (\frac{1}{2}, 2)$