Functions

A function is a mathematical concept that describes the relationship between two sets of numbers, known as the input and output.

In a function, each input value is associated with exactly one output value. Functions help us understand the relationship between variables.

Function Notation

Function notation is a shorter way of expressing a function and its input-output relationship. A function is typically denoted by f(x), where x represents the input value, and f(x) represents the corresponding output value.

For example, let’s look at the function f(x) = 2x + 1. When the input is x = 3, the output is f(3) = 2(3) + 1 = 7.

To evaluate a function for a specific input value, substitute the input value into the function rule and simplify the expression. For example, let’s evaluate the function f(x) = x^2 - 4x + 3 at x = 2:

f(2) = (2)^2 - 4(2) + 3

= 4 - 8 + 3 = -1

So, f(2) = -1

Inverse Functions

An inverse function reverses the input-output relationship of the original function. If the function f takes an input x and produces an output y, then its inverse function, denoted as f^{-1}(x), takes the input y and produces the output x.

To find the inverse of a function, follow these steps:

Step 1: Replace the function notation f(x) with the variable y.

Step 2: Swap the input (x) and output (y) values in the equation.

Step 3: Solve the equation for the new output variable (y).

For example, let’s find the inverse of the function f(x) = 2x + 1:

1. Replace f(x) with y:

y = 2x + 1

2. Swap the input (x) and output (y) values:

x = 2y + 1

3. Solve for the new output variable (y):

x - 1 = 2y

y = \frac{(x - 1)}{2}

So, the inverse function is f^{-1}(x) = \frac{(x - 1)}{2}.

Composite Functions

A composite function is the result of applying one function to the output of another function. If we have two functions, f(x) and g(x), the composite function denoted as (f \: g)(x) or f(g(x)) represents applying the function g first and then applying the function f to the result.

To find a composite function, follow these steps:

Step 1: Identify the functions f(x) and g(x).

Step 2: Write the composite function notation (fg)(x) or f(g(x)).

Step 3: Substitute g(x) into the function f.

For example, let’s find the composite function (fg)(x) for f(x) = 2x + 1 and g(x) = x^2:

1. Identify the functions:

f(x) = 2x + 1

g(x) = x^2

2. Write the composite function notation:

(fg)(x) = f(g(x))

3. Substitute g(x) into the function f:

(fg)(x) = f(x^2)

= 2(x^2) + 1

So, the composite function (fg)(x) = 2x^2 + 1.

Examples

Example 1:

Given the function f(x) = 3x - 7, find f(4).

To find f(4)f(4), substitute x = 4x = 4 into the function:

f(4) = 3(4) - 7f(4) = 3(4) - 7

= 12 - 7= 12 - 7

= 5= 5

So, f(4) = 5f(4) = 5.

Example 2:

Find the inverse of the function f(x) = 4x - 5.

To find the inverse function, follow these steps:

1. Replace f(x)f(x) with yy: y = 4x - 5y = 4x - 5

2. Swap xx and yy: x = 4y - 5x = 4y - 5

3. Solve for yy: 4y = x + 54y = x + 5 y = \frac{x + 5}{4}y = \frac{x + 5}{4}

So, the inverse function is f^{-1}(x) = \frac{x + 5}{4}f^{-1}(x) = \frac{x + 5}{4}.

Example 3:

Given the functions f(x) = 2x + 3 and g(x) = x^2 - 1, find (fg)(x) and (gf)(x).

1. Find (fg)(x) = f(g(x))(fg)(x) = f(g(x)):

Substitute g(x)g(x) into the function f: (fg)(x) = f(x^2 - 1)(fg)(x) = f(x^2 - 1)

= 2(x^2 - 1) + 3= 2(x^2 - 1) + 3

= 2x^2 - 2 + 3= 2x^2 - 2 + 3

= 2x^2 + 1= 2x^2 + 1

So, (fg)(x) = 2x^2 + 1(fg)(x) = 2x^2 + 1.

2. Find (gf)(x) = g(f(x))(gf)(x) = g(f(x)):

Substitute f(x) into the function g: (gf)(x) = g(2x + 3)(gf)(x) = g(2x + 3)

= (2x + 3)^2 - 1= (2x + 3)^2 - 1

= (4x^2 + 12x + 9) - 1= (4x^2 + 12x + 9) - 1

= 4x^2 + 12x + 8= 4x^2 + 12x + 8

So, (gf)(x) = 4x^2 + 12x + 8(gf)(x) = 4x^2 + 12x + 8.

Example 4:

Given the function h(x) = \frac{x^2 - 3x + 2}{x - 1}, find h(2).

To find h(2)h(2), substitute x = 2x = 2 into the function:

h(2) = \frac{(2)^2 - 3(2) + 2}{2 - 1} = \frac{4 - 6 + 2}{1} = \frac{0}{1} = 0h(2) = \frac{(2)^2 - 3(2) + 2}{2 - 1} = \frac{4 - 6 + 2}{1} = \frac{0}{1} = 0

So, h(2) = 0h(2) = 0.

Example 5:

Question: Find the inverse of the function p(x) = \frac{3x - 4}{2}.

To find the inverse function, follow these steps:

1. Replace p(x) with y: y = \frac{3x - 4}{2}y = \frac{3x - 4}{2}

2. Swap xx and yy: x = \frac{3y - 4}{2}x = \frac{3y - 4}{2}

3. Solve for y: 2x = 3y - 42x = 3y - 4

3y = 2x + 43y = 2x + 4

y = \frac{2x + 4}{3}y = \frac{2x + 4}{3}

So, the inverse function is p^{-1}(x) = \frac{2x + 4}{3}p^{-1}(x) = \frac{2x + 4}{3}.

Example 6:

Question: Given the functions m(x) = \sqrt{x} and n(x) = x^2 + 4, find (mn)(x) and (nm)(x).

1. Find (mn)(x) = m(n(x))(mn)(x) = m(n(x)):

Substitute n(x)n(x) into the function m: (mn)(x) = m(x^2 + 4)(mn)(x) = m(x^2 + 4)

= \sqrt{x^2 + 4}= \sqrt{x^2 + 4}

So, (mn)(x) = \sqrt{x^2 + 4}(mn)(x) = \sqrt{x^2 + 4}.

2. Find (nm)(x) = n(m(x))(nm)(x) = n(m(x)):

Substitute m(x)m(x) into the function n: (nm)(x) = n(\sqrt{x})(nm)(x) = n(\sqrt{x})

= (\sqrt{x})^2 + 4= (\sqrt{x})^2 + 4

= x + 4= x + 4

So, (nm)(x) = x + 4(nm)(x) = x + 4.

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