Question

Suppose f is a function that takes a real number x and performs the following steps in the order given:

Answer 1

Answer:

Expression for f(x) = $\frac{\sqrt{x + 1} - 4}{7}$

Domain in interval notation: $\:[-1,\:\infty \:)$

Step-by-step explanation:

$\textrm{Let }f(x)\text{ represent the function}$

Step 1: Add 1 to $x$  ==>   x + 1

Step 2: Take square root ==> $\sqrt{x+1}$

Step 3: Subtract 4: ==>  $\sqrt{x+1} - 4$

Step 4: Divide by 7 ==> $\frac{\sqrt{x+1} - 4}{7}$

The expression for f(x) is $\frac{\sqrt{x+1} - 4}{7}$

The domain of a function is the set of all inputs  for which the function is real and defined

$\sqrt{x+1}$ is real only for positive values of the square root

This means

$\sqrt{x+1} \ge 0$

Squaring both sides we get

$x + 1 \ge 0$

Subtracting 1 from both sides we get

$x \ge -1$

This is the lower limit for $x.$ There is no upper limit

The domain of$f(x)$ is therefore

$x \ge -1$

In interval notation it is expressed as

$[-1,\infty)$

Note the square bracket on the left and parenthesis on the right. This notation means that -1 is part of the domain but $\infty$  is not included since at $\infty$  the function is not defined