Give the three general forms for:
a) the reciprocal of a linear function
b) the reciprocal of a quadratic function
c) a rational function
Please give me in a full details for each please...
reciprocal of a linear function?
I suppose we could say:
f(x) = 1/(mx + b)
e.g. y = 1/(3x -5)
reciprocal of a quadratic function?
How about
f(x) = 1/(ax^2 + bx+ c)
rational functions could have many different forms
(there has to be an independent variable in the denominator)
e.g. f(x) = 3x/(4x + 5)
e.g. g(x) - 4x^2/( (4x+1)(x^3 + 6)(x^2+6) )
Was not aware of "three general forms" for any of those three types.
a) The reciprocal of a linear function:
A linear function can be represented in the form of y = mx + b, where m represents the slope and b represents the y-intercept.
To find the reciprocal of a linear function, we need to take the reciprocal (multiplicative inverse) of the entire equation, essentially switch the numerator and denominator of the equation. The reciprocal of a linear function can be expressed in three general forms:
1) In the form of y = 1/(mx + b):
- Start with the linear equation y = mx + b.
- Take the reciprocal of the entire equation by flipping the numerator and denominator, giving you y = 1/(mx + b).
2) In the form of x = 1/(my + b):
- Start with the linear equation y = mx + b.
- Swap the places of x and y in the equation: x = my + b.
- Take the reciprocal of the entire equation by flipping the numerator and denominator, resulting in x = 1/(my + b).
3) In the form of 1/y = mx + b:
- Start with the linear equation y = mx + b.
- Take the reciprocal of both sides of the equation, giving you 1/y = 1/(mx + b).
- Simplify the right side by flipping the expression to mx + b, yielding 1/y = mx + b.
b) The reciprocal of a quadratic function:
A quadratic function can be expressed in the general form of y = ax^2 + bx + c, where a, b, and c are constants.
To find the reciprocal of a quadratic function, we follow a similar process as with linear functions, but this time we take the reciprocal of the entire equation, including the constants. The reciprocal of a quadratic function can be written in three general forms:
1) In the form of y = 1/(ax^2 + bx + c):
- Start with the quadratic equation y = ax^2 + bx + c.
- Take the reciprocal of each term in the equation, resulting in y = 1/(ax^2 + bx + c).
2) In the form of x = 1/(ay^2 + by + c):
- Start with the quadratic equation y = ax^2 + bx + c.
- Swap the places of x and y in the equation: x = ay^2 + by + c.
- Take the reciprocal of the entire equation by flipping the numerator and denominator to x = 1/(ay^2 + by + c).
3) In the form of 1/y = ax^2 + bx + c:
- Start with the quadratic equation y = ax^2 + bx + c.
- Take the reciprocal of both sides of the equation, giving you 1/y = 1/(ax^2 + bx + c).
- Simplify the right side by flipping the expression to ax^2 + bx + c, yielding 1/y = ax^2 + bx + c.
c) A rational function:
A rational function is the ratio of two polynomial functions. It can be expressed in the form of f(x) = p(x) / q(x), where p(x) and q(x) are polynomial functions and q(x) ≠ 0.
There are a few general forms for rational functions, depending on the degree of the polynomials involved:
1) If the degree of the polynomial in the numerator is less than the degree of the polynomial in the denominator, the rational function is a proper rational function. Example: f(x) = (ax + b) / (cx^2 + dx + e).
2) If the degree of the polynomial in the numerator is equal to the degree of the polynomial in the denominator, the rational function is an improper rational function. Example: f(x) = (ax^2 + bx + c) / (dx^2 + ex + f).
3) If the degree of the polynomial in the numerator is greater than the degree of the polynomial in the denominator, the rational function can be expressed as a polynomial and a proper fractional part. Example: f(x) = (ax^3 + bx^2 + cx + d) / (ex^2 + fx + g) + (hx + i) / (ex^2 + fx + g).