Question

What is the range of the function f(x) = |x – 3| + 4?A. R: {f(x) ∈ ℝ | f(x) ≥ 4}B. R: {f(x) ∈ ℝ | f(x) ≤ 4}C. R: {f(x) ∈ ℝ | f(x) > 7}D. R: {f(x) ∈ ℝ | f(x) < 7}

Answer 1

A. R: {f(x) ∈ ℝ | f(x) ≥ 4}

Explanation

An absolute value function is a function that contains an algebraic expression within absolute value symbol,the absolute value of a number is its distance from 0 on the number line so as a distance it is always positive

$\lvert{x}\rvert=\begin{cases}x\text{ if x }\ge0 \\ -x\text{ if x}<0\end{cases}$

also, the range of a function refers to the entire set of all possible output values of the dependent variable.

hence ,let's check the outputs for this absolute value function

Step 1

given

$f(x)=\lvert{x-3}\rvert+4$

a)Since we have absolute signs, we must get only positive values by applying any positive and negative values for x in the given function. So, the range of absolute value is

$\begin{gathered} x-3+4,\text{ if x-3}\ge0 \\ and \\ -(x-3)\text{ i f x-3}<0 \end{gathered}$

let's solve the inequalities

$\begin{gathered} x\ge3 \\ -x+3<0 \\ \cap \\ -x<-3 \\ x>3 \\ \end{gathered}$

so, the value in the absolute value sign will be alwasys greater or equa

than 0 ( we have to add the 4 )

so

$\begin{gathered} f(x)=(\text{ greater tahn 0\rparen+4} \\ \\ it\text{ means the function will be always greater or equal than 4} \end{gathered}$

so, the range of the function is all the numbers greater or equal than 4,

therefore, in set notation the answer is

A. R: {f(x) ∈ ℝ | f(x) ≥ 4}

I hope this helps you