Single Variable Functions Domains

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Its All about Pj Problem Strings - 7 Spaces Of Interest and their associated Basic Sequences; 7 Pj Problems of Interest (PPI) and their Alleles (A)

Single Variable Functions - Domains

A function f, is a one-one mapping from its domain to its range. A single variable function f is often expressed in association with its independent and dependent variables as:
f(x) = y = an expression of x (other letters can be used). In this representation, x is the independent variable (it constitutes the domain of f) and y is the dependent variable (it constitutes the range of f

1. What is meant by:
(a) The domain of a function?
(b) The range of a function?
Ans:1(a) The values that can be assigned to the independent variable of a function.
1(b) The values of the dependent variable evaluated at each value of the independent variable. For example, f(x) at x = 2 is the value f(2) and is a value in the range of the function f.

2. A curve has infinitely many points at x = -1. Is this curve the graph of a function?
Ans: No. There must be a one-one correspondence between the domain and range of a function.

3. Determine the domains of the following functions:
(a) f(x) = x4/(x2 + x - 6)
(b) f(u) = (u -1)1/3
Ans: (a) Since x2 + x - 6 cannot be 0, function is not defined for values of x for which x2 + x - 6 = (x + 3)(x - 2) = 0; i.e. for x= -3 or 2. Therefore domain of function is {all x ∈ R | x ≠ -3; 2}. Where R = all real numbers.
(b) Function is defined for every u, since every real number has a cube root. So the domain is {all u ∈ R}.

4. What part of the real number line is excluded from the domain of the following function:
f(x) = x/|x|
Ans: f(x) = x/x = 1 for x > 0; f(x) = x/-x = -1 for x< 0. undefined for x = 0.
So the domain is {all x ∈ R | x ≠ 0;}. So x = 0 is excluded.

5. What is the least number in the domain of the following function:
f(x) = 3 - 2x
Ans: The graph of the function intercepts the y-axis at 3 and the x- axis at 3/2, then it tends to infinity at both ends. Domain is {all x ∈ R}. The least number in this domain is -∞.

6. What part of the domain of the following function refers to the horizontal line in the graph of the following function:
G(x) = |x| + x
Ans: Function evaluates to G(x) = 2x; x ≥ 0; G(x) = 0; x < 0. Domain is {all x ∈ R}. The part of the graph for x < 0 is horizontal.

7. The perimeter of a rectangular area is 20.
(a) What is the domain of the area when expressed as a function of its length.
(b) What is the domain of the area when expressed as a function of its length and the length is restricted to be larger than the width.
Ans: (a) Let the width = W and the length = L. Then 2W + 2L = 20
So, W = (20 - 2L)/2. Then Area = [(20 - 2L)/2]L = 10L - L2. Assuming positive length. Domain of area is 0 < L < 10.
Restriction requires L > W. So L > (20 - 2L)/2. Therefore domain of area is 5 < L < 10.

8. The volume of a cubic box is 2. The surface area is S = x2 + 4xh (where h is the height of the box). What is the domain of the function that expresses the ratio of the volume of the box to its surface area.
Ans: Let a side of the box be x. Then Volume of cubic box, V = 2 = x3. Surface area S = 5x2. Ratio V:S = f(x) = x3/5x2 = x/5. Domain is x > 0.

9. Given that f(x) is an even function and the point (5,3) is on its graph. Indicate a point that must also be on the graph.
Ans: An even function is symmetric with respect to the y-axis. Since the point (5,3) is on the graph, the point (-5,3) must also be on the graph.

10. Given that f(x) is an odd function and the point (5,3) is on the graph. Indicate a point that must also be on the graph.
Ans: An odd function is symmetric with respect to the origin. Since the point (5,3) is on the graph, the point (-5,-3) must also be on the graph.

Single Variable Functions: Domains
Single Variable Functions: Limits. Differentiability And Continuity
Single Variable Functions The Derivative As Rate Of Change
Conics
Vector Spaces
Real Numbers