MATEMATİK I

Substituting x for -1, we find 5.(-1²) + 10.(-1) + 7 = 5 . 1 + (-10) + 7 = 5 -10 + 7 = 2.

It is clear that this function has an inverse. Hence, for finding f^-1 (x), we write y = 3x + 5. What we find is x = (5 - y) / 3. Then, f^-1 (x) = (5 - y) / 3. Substituting 2 with x, we find 1.

(f ∘ g)(2) equals to f ( g(2) ). g(2), substituting 2 with x in the g function is -2 +3 = 1. Then, (f ∘ g)(2) = f (1) = 3 . 1 -7 = 3 - 7 = -4.

(g ∘f)(-1) = g ( f(-1) ) = g (-4) = -66

This function makes sense if its denominator is not zero, i.e. the condition of 2x - 10 ≠ 0 should be satisfied. Hence, if x ≠ 5, this condition is satisfied.  Therefore, the largest domain of the function is R \ {5}.

Constant functions map every element from their domain to the same element of the range. We may therefore define a constant function for every element in the range. Since there are 9 different elements in the range we may write 9 different constant functions.

Let A and B be two sets different from the empty set. A correspondence assigning each element of the set A, one and only one element of the set B is called a function from the set A into the set B. Functions are generally denoted by the lower case letters such as f, g, h. In this case a function defined from the set A to the set B is denoted by

                                    f:  or . 

In this context, the set A is called the domain or the departure set of the function, and the set B is called the range or terminal set. For a given function f and an element a from its domain A, the element corresponding to a under the rule f  is called the image of the element a, and is denoted by f (a).

As also understood from the information given, all of the statements regarding to functions in the options are correct, so the correct answer is E.

Consider the function that assigns to each real number three times itself and four. If we denote this real number by x, and the function by f we may express it as

It is customary to also denote this function by y = f (x).

In the expression y = f (x), x is called the independent variable, whereas y is called the dependent variable.

In the expression y is called the dependent variable.

In the equation y = f (x)=3x+4, the dependent variable y is a function of the independent variable x.

As also understood from the information given, the correct answer is E. All of the statements in the options are correct.

Substituting 3 for x we find f (3) = 5.3 - 4 = 15 – 4 = 11, so the correct answer is A.

From the equation 3x+6=y we find x = f ^-1(y) = . Thus f ^-1(x) = , so the correct answer is B.

If a rule defining a function is given but the domain has not been specified explicitly, then the largest set which makes the rule meaningful is understood.

This set is denoted by D_f  for the rule y=f (x) and is called the natural domain of the function.

The function given by the rule f(x) =makes sense if its denominator is not zero, i.e. x–1≠0 should be satisfied.

As also understood from the information given, the correct answer is E.

Consider the function f: . 

For every x₁, x₂A if  x₁ ≠ x₂ implies

f(x₁) ≠ f(x₂)

then the function f is called one-to-one (or injective), so the correct answer is A.

Although the range and the image of a function are two concepts that need to be distinguished, we still know that they are closely related. The image is always a subset of the range. Yet, there are functions for which the range and the image sets are exactly the same. Thus, given a function f : if the image is equal to the range, i.e. f (A)=B the function f  is called surjective (onto). Equivalently, for every element b of the set B if there can be found an element a of A such that f (a)=b, then f  is called onto.

If a function which is both one-to-one and onto is called a bijection.

Constant function and identity function are the concepts related to types of functions.

A function assigning each element from its domain a single element in its range is called a constant function.

Given A ≠, a function defined on the set A and assigning every element of A to itself is called the identity function.

Consider the function . For every x₁, x₂A if  x₁ ≠ x₂ implies

f(x₁) ≠ f(x₂) then the function f is called one-to-one (or injective).

Given a function  if the image is equal to the range, i.e. f (A)=B the function f  is called surjective (onto).

If a function which is both one-to-one and onto is called a bijection.

Constant function and identity function are the concepts related to types of functions. A function assigning each element from its domain a single element in its range is called a constant function.

Given A ≠  a function defined on the set A and assigning every element of A to itself is called the identity function.

As also understood from the information all of the statements regarding to properties and types of functions in the options are correct, so the correct answer is E.

Functions, which are represented by different formulas on different subsets of its domain are called piecewise defined functions, so the correct answer is C. Definitions for the other concepts regarding to functions in the other options are as follows:

Constant function and identity function are the concepts related to types of functions. A function assigning each element from its domain a single element in its range is called a constant function.

Given A ≠ , a function defined on the set A and assigning every element of A to itself is called the identity function.

Given a function  if the image is equal to the range, i.e. f (A)=B the function f  is called surjective (onto).

If a function which is both one-to-one and onto is called a bijection.

The inequality  should be satisfied. Hence, D_f = (-∞, 3], so the correct answer is A.

Now is the time to plot the graph of functions in order to make them visible to the eye. We will talk about the Cartesian coordinate system, a tool which enables us to view the graphs of functions.

Let us get acquainted with the Cartesian coordinate system. Two real lines intersecting perpendicularly at both their zeros constitute the Cartesian coordinate system. In this system, the horizontal real line is called the x-axis, or abscissa, and the vertical real line is called the y-axis, or ordinate. We call the point of intersection of the number lines as the origin.

As also understood from the information given, the correct answer is E. All of the statements regarding to graphs of functions in the options are correct.

(4f –2g)(x) = 4(x-2) - 2(2x-9) = 4x - 8 - 4x + 18 = 10

In order for the square root to make sense, the argument (the terms inside the square root) should not be negative. (2x - 12)^1/2 is the square root of 2x - 12. Therefore, the inequality 2x–12 ≥ 0 should be satisfied. This means that x ≥ 6 must be satisfied. Hence, the largest domain of definition of f is [6, ∞).

Since g cannot be zero, f / g is meaningful, and the answer can be calculated. 

The answer is, then, (-2+6) / ((-1)² +3) = 4 / 4 ) = 1

4(4 + 9) - 3(12-6) = 52 - 18 = 34