Question

The graph of f(x) = |x| is translated 6 units to the right and 2 units up to form a new function. Which statement about the range of both functions is true? The range is the same for both functions: {y | y is a real number}. The range is the same for both functions: {y | y > 0}. The range changes from {y | y > 0} to {y | y > 2}. The range changes from {y | y > 0} to {y | y > 6}.

Answer 1

Step-by-step explanation:

Question :-

The graph of f(x) = |x| is translated 6 units to the right and 2 units up to form a new function. Which statement about the range of both functions is true ?

(A) The range is the same for both functions: {y | y is a real number}.

(B) The range is the same for both functions: {y | y > 0}.

(C) The range changes from {y | y > 0} to {y | y > 2}.

(D) The range changes from {y | y > 0} to {y | y > 6}.

Solution :-

We have f(x) = |x|

The vertex of this function is the point (0,0)

The range of f(x) is the interval [0, ∞)

If f(x) translated 6 units to the right and two units up to form a new function g(x)

Then the rule of translation of f(x) to g(x) should be

${\bold{\sf{⟹ \: \: \: \: \: (x, \: y) \: = \: (x \: + \: 6, \: y \: + 2 \: )}}}$

The new vertex is

${\bold{\sf{⟹ \: \: \: \: \: (0, \: 0) \: = \: (0 \: + \: 6, \: 0 \: + \: 2)}}}$

${\bold{\sf{⟹ \: \: \: \: \: (0, \: 0) \: = \: 6, \: 2}}}$

So the new function equation is

${\bold{\sf{⟹ \: \: \: \: \: g(x) \: = \: |x - 6 | \: + \: 2}}}$

The range of g(x) is the interval [2, ∞)

Therefore the correct statement is (C)

The range changes from {y | y > 0} to {y | y > 2}.