F(x)={ ln (-x) if x<0 ln x if x>0 . |...

Key Concepts

Transformation to Absolute Value

A common transformation in piecewise-defined logarithmic functions involves rewriting expressions such as ln(-x) for x < 0 as ln|x|. This re-expression highlights that the natural logarithm is applied to the absolute value of x, ensuring the input is positive. It is a useful technique for simplifying analysis and linking behavior across subdomains.

Natural Logarithm Function

The natural logarithm function, denoted as ln(x), is defined only for positive values of x. This foundational concept in calculus and real analysis means that any expression involving ln(x) must ensure the argument is greater than zero to be valid.

Domain Determination

Finding the correct domain of a function is a key analysis step, especially when dealing with logarithms. For an expression like ln(-x) when x is negative, the negative sign ensures that the inner expression becomes positive, satisfying the logarithm's requirement. This technique helps in articulating the domain of each piece in a piecewise function.

Piecewise-defined Functions

Piecewise functions have different rules or formulas for different parts of their domains. Understanding how to analyze each piece separately—considering their individual domains, properties, and how they might connect—is crucial for a complete understanding of the function's behavior.

Transcript

00:01 That has been given to us is fx is equal to ln minus x if x is less than zero and it will be equal to ln x if x is greater than zero fine this is the question given to us you've been asked to plot a graph for the given function so we all know the basic graph for ln x that is this red one isn't it and ln minus x is just obtained by reflecting the graph of lnx on the y axis.

00:35 So this will be the reflection of the parent function that is lnx, isn't it? and on the y -axis, if i'm reflecting it, i'll get that black curve only, no? so that will be the graph of minus lnx.

00:49 So this will be the required graph over here.

00:53 Now, apart from that, you've been asked to just determine a few things, like the basic thing will be domain that you have to determine over here from the graph itself.

01:03 But before beginning, let us first discuss what is meant by this term domain.

01:09 So simply, my dear friends, domain you can say refers to the set of possible input values, isn't it? these are just a set of possible input values.

01:23 Right.

01:24 And the domain of a graph consists of all the input values that are shown on the xx.

01:29 Axis so if i say if i'm saying all the possible values that are there on the x axis then what will be the domain for the given graph over here simply we can easily say the domain is starting from minus infinity till zero but zero is not getting included so i'll use a normal bracket over here union again starting from zero till positive infinity right like this one or i can also write that in the other notation as x such that x does not belongs to zero that means it will include the domain will include all the values on the x axis except zero and that is clear isn't it you can see that the zero over here that is not getting included in the graph isn't it so simply the domain is the values of x such that x is not equal to zero move on to the next one, you've been asked to determine the range as well.

02:35 Range that is also simple.

02:37 It is just a set of possible output values that are on the y -axis this time.

02:43 See, domain was the possible input values and range is the set of possible output values.

02:48 That's the basic difference between domain and range.

02:51 And yeah, the basic thing also, domain is determined by the values that are on the x -axis, whereas the range is determined by the values that are on the x -axis, whereas the range is determined by the values that are shown on the y -axis...

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