I NEED HELP WITH QUESTION PLEASE!!
What is the inverse of f(x)
(x) = (x-4)² for x ≥ 4?

I NEED HELP WITH QUESTION PLEASE!!What Is The Inverse Of F(x)(x) = (x-4) For X 4?

Answers

Answer 1

Answer:

4th choice [tex]\bold{f^{-1}(x) = \sqrt{x} + 4}[/tex]

Step-by-step explanation:

Definition of the inverse of a function

A function g is the inverse of a function f if whenever y=f(x) then x=g(y). In other words, applying f and then g is the same thing as doing nothing. We can write this in terms of the composition of f and g as g(f(x))=x. The domain of f becomes the range of g and the range of f becomes the domain of g

To solve for the inverse of the function [tex]f(x) =\left(x-4\right)^2[/tex]

Let [tex]y=\left(x-4\right)^2[/tex]

[tex]\mathrm{Replace}\:x\:\mathrm{with}\:y[/tex] [tex]\text{ and replace }\:y\:\mathrm{with}\:x[/tex]

[tex]x=\left(y-4\right)^2[/tex]

Switch sides
[tex]\left(y-4\right)^2=x[/tex]

Take square roots on both sides
[tex]y-4=\pm\sqrt{x}[/tex]

Add 4 on both sides to solve for y

[tex]y = \pm\sqrt{x} + 4[/tex]

We have two solutions

[tex]y=\sqrt{x}+4,\:y=-\sqrt{x}+4[/tex]

To determine which one of these to be chosen not that in the given choices we can eliminate the first two since x cannot be negative

The third choice can also be eliminated since
[tex]-\sqrt{x} + 4[/tex] is a decreasing function for [tex]x \ge 0[/tex]

So the last answer choice is correct and the inverse of[tex]f(x) = (x-4)^2[/tex]

is given by [tex]f^{-1}(x) = \sqrt{x} + 4[/tex]

Answer:4th choice [tex]\bold{f^{-1}(x) = \sqrt{x} + 4}[/tex]

Note

Domain of (x-4)² is [4, ∞) since x ≥ 4 and (x-4)² cannot be negative

Range of (x-4)² is [0, ∞)

Domain of [tex]\sqrt{x}\:+\:4[/tex] is [0, ∞)

Range of [tex]\sqrt{x}\:+\:4[/tex] is [4, ∞)

so indeed the domain of (x-4)² has become the range of [tex]\sqrt{x}\:+\:4[/tex] and the range of (x-4)² has become the domain of [tex]\sqrt{x}\:+\:4[/tex]

Related Questions

select two points that lie on the same line throught (0,0), then select an equation that describes the relationship between the coordinates x and y

Answers

Two collinear points on a line are given in the table below:

x, y

0 0 (0,0)

2 1 (2,1)

Select the points that lie on the line. and the relationship between the coordinates x and y.

(4,2)

(4, 3)

(7,2)

4/8, 2/8

Answer:

(4,2) and ([tex]\frac{4}{8}[/tex],[tex]\frac{2}{8}[/tex]) are lie on the line, and y = [tex]\frac{1}{2}[/tex] x , is showing the relationship of x and y coordinates of given line.

Given that

The co-ordinates of line is (0,0) and (2,1).

To find

The points that lie on the line, and The relationship between the coordinates x and y.

So, according to the question

We have,

The X and Y co-ordinates of line (0,0) and (2,1).

For finding any point that lie on line, first we have to find the slope of given line,

So, we know that

slope (m) = [tex]\frac{y_2-y_1}{x_2-x_1}[/tex]

Where,

x₁ = 0 and x₂ = 2

y₁ = 0 and y₂ = 1

Now, putting the values, we will get

(m) = [tex]\frac{1-0}{2-0}[/tex] = [tex]\frac{1}{2}[/tex]

m = [tex]\frac{1}{2}[/tex]

We know that, the equation of line

y-y₁ = m(x- x₁ )

y-0 = [tex]\frac{1}{2}[/tex] (x- 0)