This table shows the input and output values for an exponential function f(x) . what is the ratio of outputs for any two inputs that are one value apart? 1/2 4 2 1/8

The correct answer is:  [B]:  "4 " .

Explanation: 

Refer to the table (provided within the actual question).

Note that the "inputs" ; or "x-values" ; are all listed in "chronological order" ; and are all "one (1) unit apart. and range from:  "x = -3"  to "x = 3" .

When "x = 0" ; the "output" ; or "f(x)"  is "1/4" . 
When "x = 1"  ; the "output" ; or "f(x)" is:  "1" .

So; the ratio of these two "outputs"  is:  "¼ : 1"  ; or, write as:
  " (¼) / 1 " ;   and note that:  " (¼) / 1 =  (¼) ÷ 1 = ¼.

However; note that: "1/4" ; or  "1:4" is NOT among the [answer choices given].

However, the ratio of the 2 (two) corresponding "outputs"; chronologically,
going from when "x = 1" ; to "x = 0" ;  is:  "1 : ¼" ; or;  write as:  "1 / (¼)" ; 

And note that:  "1 / (¼)"  =  " 1 ÷ (¼) " = 1 * (4/1) = 1 * 4 = "4" .

This corresponds to:  Answer choice: [B]:  "4" .

Let us further confirm that this answer is correct:

When x = 3;  the "output" is:  "16" .

When x = 2;  the "output" is:  "4" .

The ratio:  "16/4 = ?  4 ? " ;    →  Yes!

When x = 2;  the "output" is:  "4" .

When x = 1;   the "output" is:  "1" .

The ratio:  "4/1  = ?  4 ? " ;    →  Yes!

When x = 1;  the "output" is:  "1" .

When x = 0; the "output" is:  "(¼)" .

The ratio:  "1 / (¼)  = ?  4 ? " ;  

   →  "1 / (¼)"  =  " 1 ÷ (¼) " = 1 * (4/1) = 1 * 4 = "4" . YES!

When  x = 0; the "output" is:  "(¼)" .

When  x = -1; the "output" is:  "(¹/₁₆)" .

The ratio:  "(¼) / (¹/₁₆)  = ?  4 " ? ;  
  
   →  "(¼) / (¹/₁₆) = "(¼) ÷ (¹/₁₆) " =  "(¼) * (¹⁶/₁) = (1*16) / (4*1) = 16/4 = "4" .  Yes!

When  x = -1; the "output" is:  "(¹/₁₆)" .

When  x = -2; the "output" is:  "(¹/₆₄)" .

The ratio:  "(¹/₁₆) / (¹/₆₄)  = ?  4 " ? ;  
  
   →  "(¹/₁₆) / (¹/₆₄)  = "(¹/₁₆) ÷ (¹/₆₄)" = "(¹/₁₆) * (⁶⁴/₁)" = (1*64) / (16*1) = 64/16 = "4" . Yes!

When  x = -2; the "output" is:  "(¹/₆₄)" .

When  x = -3; the "output" is:  "(¹/₂₅₆)" ,

The ratio:  "(¹/₆₄)/(¹/₂₅₆)  = ?  4 " ? ;  
  
 → "(¹/₆₄) / (¹/₂₅₆)"  ;
 
       =   " (¹/₆₄) ÷ (¹/₂₅₆)" = " (¹/₆₄) * (²⁵⁶/₁) " = (1*256) / (64*1) = 256/164  =  "4 " . Yes!

  → So; as calculated; the ratio is:  "4" ;  which is:

  →  Answer choice:  [B]:  "4" .