Which of the following describes the functions of the relationship between two functions of the following
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If we in the following equation y=x+7 assigns a value to x, the equation will give us a value for y. Example $$y=x+7$$ $$if\; x=2\; then$$ $$y=2+7=9$$ If we would have assigned a different value for x, the equation would have given us another value for y. We could instead have assigned a value for y and solved the equation
to find the matching value of x. In our equation y=x+7, we have two variables, x and y. The variable which we assign the value we call the independent variable, and the other variable is the dependent variable, since it value depends on the independent variable. In our example above, x is the independent variable and y is the dependent variable. A function is an equation that has only one answer for y for every x. A function assigns exactly one output to each input of a specified
type. It is common to name a function either f(x) or g(x) instead of y. f(2) means that we should find the value of our function when x equals 2. Example $$f(x)=x+7$$ $$if\; x=2\; then$$ $$f(2)=2+7=9$$ A function is linear if it can be defined by $$f(x)=mx+b$$ f(x) is the value of the function. This form is called the slope-intercept form. If m, the slope, is negative the functions value decreases with an increasing x and the opposite if we have a positive slope. An equation such as y=x+7 is linear and there are an infinite number of ordered pairs of x and y that satisfy the equation.  The slope, m, is here 1 and our b (y-intercept) is 7. $$m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$$ $$x_{2}\neq x_{1}$$ If two linear equations are given the same slope it means that they are parallel and if the product of two slopes m1*m2=-1 the two linear equations are said to be perpendicular. Video lessonIf x is -1 what is the value for f(x) when f(x)=3x+5? Summary Read a brief summary of this topicfunction, in mathematics, an expression, rule, or law that defines a relationship between one variable (the independent variable) and another variable (the dependent variable). Functions are ubiquitous in mathematics and are essential for formulating physical relationships in the sciences. The modern definition of function was first given in 1837 by the German mathematician Peter Dirichlet:
This relationship is commonly symbolized as y = f(x)—which is said “f of x”—and y and x are related such that for every x, there is a unique value of y. That is, f(x) can not have more than one value for the same x. To use the language of set theory, a function relates an element x to an element f(x) in another set. The set of values of x is called the domain of the function, and the set of values of f(x) generated by the values in the domain is called the range of the function. In addition to f(x), other abbreviated symbols such as g(x) and P(x) are often used to represent functions of the independent variable x, especially when the nature of the function is unknown or unspecified. Common functionsMany widely used mathematical formulas are expressions of known functions. For example, the formula for the area of a circle, A = πr2, gives the dependent variable A (the area) as a function of the independent variable r (the radius). Functions involving more than two variables (called multivariable or multivariate functions) also are common in mathematics, as can be seen in the formula for the area of a triangle, A = bh/2, which defines A as a function of both b (base) and h (height). In these examples, physical constraints force the independent variables to be positive numbers. When the independent variables are also allowed to take on negative values—thus, any real number—the functions are known as real-valued functions. Britannica Quiz Numbers and Mathematics A-B-C, 1-2-3… If you consider that counting numbers is like reciting the alphabet, test how fluent you are in the language of mathematics in this quiz. The formula for the area of a circle is an example of a polynomial function. The general form for such functions is P(x) = a0 + a1x + a2x2+⋯+ anxn, where the coefficients (a0, a1, a2,…, an) are given, x can be any real number, and all the powers of x are counting numbers (1, 2, 3,…). (When the powers of x can be any real number, the result is known as an algebraic function.) Polynomial functions have been studied since the earliest times because of their versatility—practically any relationship involving real numbers can be closely approximated by a polynomial function. Polynomial functions are characterized by the highest power of the independent variable. Special names are commonly used for such powers from one to five—linear, quadratic, cubic, quartic, and quintic for the highest powers being 1, 2, 3, 4, and 5, respectively. Polynomial functions may be given geometric representation by means of analytic geometry. The independent variable x is plotted along the x-axis (a horizontal line), and the dependent variable y is plotted along the y-axis (a vertical line). When the graph of a relation between x and y is plotted in the x-y plane, the relation is a function if a vertical line always passes through only one point of the graphed curve; that is, there would be only one point f(x) corresponding to each x, which is the definition of a function. The graph of the function then consists of the points with coordinates (x, y) where y = f(x). For example, the graph of the cubic equation f(x) = x3 − 3x + 2 is shown in the figure. Get a Britannica Premium subscription and gain access to exclusive content. Subscribe Now Another common type of function that has been studied since antiquity is the trigonometric functions, such as sin x and cos x, where x is the measure of an angle (see figure). Because of their periodic nature, trigonometric functions are often used to model behaviour that repeats, or “cycles.” The exponential function is a relation of the form y = ax, with the independent variable x ranging over the entire real number line as the exponent of a positive number a. Probably the most important of the exponential functions is y = ex, sometimes written y = exp (x), in which e (2.7182818…) is the base of the natural system of logarithms (ln). By definition x is a logarithm, and there is thus a logarithmic function that is the inverse of the exponential function. Specifically, if y = ex, then x = ln y. Nonalgebraic functions, such as exponential and trigonometric functions, are also known as transcendental functions. Complex functionsPractical applications of functions whose variables are complex numbers are not so easy to illustrate, but they are nevertheless very extensive. They occur, for example, in electrical engineering and aerodynamics. If the complex variable is represented in the form z = x + iy, where i is the imaginary unit (the square root of −1) and x and y are real variables (see figure), it is possible to split the complex function into real and imaginary parts: f(z) = P(x, y) + iQ(x, y). What is the relationship between the functions?Definition of Relation and Function in Maths
Functions- The relation that defines the set of inputs to the set of outputs is called the functions. In function, each input in the set X has exactly one output in the set Y. Note: All functions are relations but all relations are not functions.
What are the two types of function relationships?Bijective Function - A function that is both one-to-one and onto function is called a bijective function. Constant Function - The constant function is of the form f(x) = K, where K is a real number. For the different values of the domain(x value), the same range value of K is obtained for a constant function.
How do you describe the relationship between two quantities?Two quantities have a proportional relationship if they can be expressed in the general form y = kx, where k is the constant of proportionality. In other words, these quantities always maintain the same ratio. That is, when you divide any pair of the two values, you always get the same number k.
What is the relationship between a function and its roots?When the function has one real root, its graph is tangent to the x-axis. When it has two real roots, the vertex is halfway between the two roots. When there are no real roots, as happens in the case presented here, the complex roots still give us the x-coordinate of the vertex.
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