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Answer:
Expression for f(x) = [tex]\frac{\sqrt{x + 1} - 4}{7}[/tex]
Domain in interval notation: [tex]\:[-1,\:\infty \:)[/tex]
Step-by-step explanation:
[tex]\textrm{Let }f(x)\text{ represent the function}[/tex]
Step 1: Add 1 to [tex]x[/tex] ==> x + 1
Step 2: Take square root ==> [tex]\sqrt{x+1}[/tex]
Step 3: Subtract 4: ==> [tex]\sqrt{x+1} - 4[/tex]
Step 4: Divide by 7 ==> [tex]\frac{\sqrt{x+1} - 4}{7}[/tex]
The expression for f(x) is [tex]\frac{\sqrt{x+1} - 4}{7}[/tex]
The domain of a function is the set of all inputs for which the function is real and defined
[tex]\sqrt{x+1}[/tex] is real only for positive values of the square root
This means
[tex]\sqrt{x+1} \ge 0[/tex]
Squaring both sides we get
[tex]x + 1 \ge 0[/tex]
Subtracting 1 from both sides we get
[tex]x \ge -1[/tex]
This is the lower limit for [tex]x.[/tex] There is no upper limit
The domain of[tex]f(x)[/tex] is therefore
[tex]x \ge -1[/tex]
In interval notation it is expressed as
[tex][-1,\infty)[/tex]
Note the square bracket on the left and parenthesis on the right. This notation means that -1 is part of the domain but [tex]\infty[/tex] is not included since at [tex]\infty[/tex] the function is not defined