Question

1. Understanding and drawing graphs of Basic functionsand checking the function as one one and onto with respect to their domain and range using graph of the functions. i) |x|, |x|+1, |x|-1, |x-1| |x+1| etc ii) [x], greatest integer function. iii) 2x, 3x, x+1, x-1, 2x-3, etc iv) x2, x3, x2-1, x2+1, x3+1…etc v) sinx, cosx, tanx, secx, cosecx, cotx.

Answer 1

Answer:

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Answer 2

sets are one-one or onto are given below

Given,

i) |x|, |x|+1, |x|-1, |x-1| |x+1| etc

ii) [x], greatest integer function.

iii) 2x, 3x, x+1, x-1, 2x-3, etc

iv) x2, x3, x2-1, x2+1, x3+1…etc

v) sinx, cosx, tanx, secx, cosecx, cotx.

To Find,

Find if the given sets are one-one or onto

Solution,

• Every element in the domain of a function is said to be one-one when there is only one value in the range for each element. In order to qualify as a one-one function, a function can only have one value; if it has two solutions, it is not.
• If a function's intervals cover all values in its range, it is said to be onto. One-oneness and ontoness are simultaneous states of a function.
• In many situations, modulous functions are not one-one, and the domain of a modulus function is always positive real numbers.

Think about the first function, The given function has a range of (-infinite,+infinite), a domain of real numbers, and each value of x has a single value.

The function that is supplied is therefore both one-one and onto.

|x|, The domain of the provided function is positive real numbers, and its range is (-infinite,+infinite).

Because the graph crosses at two locations, the supplied function is not one-one. Given that every point in the range is covered, it is an onto function.

Ixl - 1 The given function has a range of (-infinite,+infinite), a domain of Positive real numbers, and is an onto function because it covers every point in the range.

Ix+1| The domain of the provided function is positive real numbers, and the given function has a range of (-infinite,+infinite).

Due to the graph's two sites of intersection, the given function is not one-one. All of the points in the range are covered, making it an onto function.

Ix-1| The domain of the provided function is positive real numbers, and the given function has a range of (-infinite,+infinite).

Due to the graph's two sites of intersection, the given function is not one-one. All of the points in the range are covered, making it an onto function.

greatest integer function: x stands for the largest integer function for any real function. The function subtracts the real number from the real number and rounds it off to the next lowest integer. Also known as the Floor Function, this function has several uses.

2x: All real numbers, with the exception of cases when the expression is undef, fall inside the expression's domain. The expression is ambiguous in this situation since there is no real number.

(-∞,∞) is a notation for intervals.

3x: All real numbers, with the exception of cases when the expression is undef, fall inside the expression's domain. The expression is ambiguous in this situation since there is no real number.

l(-∞,∞) is a notation for intervals.

Sinx: The graph of y=sin(x) has the shape of a wave that forever oscillates between -1 and 1, repeating every 2 units. Therefore, the range of sin(x) is [-1,1] and the domain is all real integers.

Cosx: All real numbers fall inside the domain of the function f(x) = cosx, but its range is 1 cosx 1. Whether an angle is measured in degrees or radians will affect the cosine function's values.

tanx: The values where cos(x) equals 0 (i.e., the values 2+n for all integers n) are not included in the domain of the function y=tan(x)). The entire real number range comprises the tangent function.

Secx will fall within the R- range (-1,1). Sec x can never be located in that region since cos x lies in the range of -1 to 1. In regions where sin x is zero, cosec x will not be defined. Cosec x's domain will therefore be R-n, where n=I.

Cosec X's operating range will be R- (-1,1). The range of -1 to 1 is where sin x sits, hence cosec x can never be found there.

Cotx: The function's domain is either y1 or y1. The cotangent function graph appears as follows: All real numbers fall inside the domain of the function y=cot(x)=cos(x)sin(x), with the exception of those where sin(x) equals 0, or the values n for all integers n. The function's range includes only real values.

Hence, these are the sets are one-one or onto

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