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A curve with arrows at both the ends facing upward is plotted on a coordinate plane with x and y axes ranging from negative 10 to 10 in unit increments.

Determine the range of the exponential function f(x)=3(12)x−6
.

(1 point)
Responses

y>6
y greater than 6

y>−6
y greater than negative 6

y<−6
y less than negative 6

y≥−6
y greater-than-or-equal-to negative 6
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1 answer

To analyze the range of the function \( f(x) = 3(12)^x - 6 \), we start with understanding the components of the function.

  1. Exponential Function Properties: The term \( (12)^x \) is an exponential function that is always positive for all real values of \( x \). Thus, \( (12)^x > 0 \).

  2. Multiplying by 3: When multiplying by 3, the function becomes \( 3(12)^x > 0 \). Therefore, \( 3(12)^x \) will also always be greater than 0.

  3. Subtracting 6: When we subtract 6 from \( 3(12)^x \), we adjust the entire range downwards by 6: \[ f(x) = 3(12)^x - 6 > 0 - 6 = -6 \] Thus, \( f(x) > -6 \).

From this analysis, we can conclude that as \( x \) approaches negative infinity, \( f(x) \) approaches -6 but never actually reaches it. As \( x \) approaches positive infinity, \( f(x) \) grows without bound.

Thus, the range of the function is: \[ f(x) > -6 \]

Given the options, the correct response is:

y > −6 (y greater than negative 6).