Chapter 1.2, Problem 1E

Solution

To find: Whether the formula determines as a function of $x$ and explain.

  $y=\sqrt{x-4}$ is a function of $x$ .

Given Information:

Equation, $y=\sqrt{x-4}$ .

Concept used:

The function is defined as a special type relation from a set $A$ to $B$ such that for every element $x$ of the set $A$ , there exist only one image $y$ in set $B$

Can be written as follow

  $f\left(x\right)=y$

Calculation:

Since, negative value of $x-4$ does not map the range to the real value

Hence the function is defined for $x-4\ge 0$

  $x-4\ge 0$

Therefore, if the function is defined for every value of $x\ge 4$ , there is a value of $y$ .

Now, observe that $\forall x\in ℝ|x\ge 4$ There exist a non-negative value of $y\in ℝ|y\ge 0$

Thus, from the definition of function mentioned there must exist one and only image

So, the equation is function

Where domain: $\left\{x:ℝ,x\ge 4\right\}$

And Range: $\left\{y:ℝ,y>0\right\}$ .

Therefore, $y=\sqrt{x-4}$ is a function