Question

Given the following absolute value function sketch the graph of the function and find the domain and range.

ƒ(x) = |x + 3| - 1

Answer 1

Find the Inverse Function f(x)=x^3-1

f(x)=x3−1f(x)=x3-1

Replace f(x)f(x) with yy.

y=x3−1y=x3-1

Interchange the variables.

x=y3−1x=y3-1

Solve for yy.


Move −1-1 to the right side of the equation by subtracting −1-1 from both sides of the equation.

y3=1+xy3=1+x

Take the cube root of both sides of the equation to eliminate the exponent on the left side.

y=3√1+xy=1+x3

Reorder 11 and xx.

y=3√x+1y=x+13

Solve for yy and replace with f−1(x)f-1(x).


Replace the yy with f−1(x)f-1(x) to show the final answer.

f−1(x)=3√x+1f-1(x)=x+13

Set up the composite result function.

f(g(x))f(g(x))

Evaluate f(g(x))f(g(x)) by substituting in the value of gg into ff.

(3√x+1)3−1(x+13)3-1

Simplify each term.


f(3√x+1)=x+1−1f(x+13)=x+1-1

Simplify by subtracting numbers.


f(3√x+1)=xf(x+13)=x

Since f(g(x))=xf(g(x))=x, f−1(x)=3√x+1f-1(x)=x+13 is the inverse of f(x)=x3−1f(x)=x3-1.

f−1(x)=3√x+1

i hope this helped.

Answer 2

Explanation:

Given is an absolute value function as

`f(x) = |x+3|-1`

This can be split for values of x <-3 and values of x >=-3

Since |x+3| cannot be negative minimum value is 0

So f(x) has minimum value as -1

There is no limit for max value

So range is [-1,∞)

Domain is all real values as x can take any value.

Graph is shown in the attachment file