Question text The exponential function [math: f(x)=66x+4+1]\displaystyle f(x)={6^{6\,x+4}+1} has been translated [math: 5]{5} units in the positive direction of the [math: x]-axis (from [math: y]-axis). (Note: Type inf for [math: ∞]\infty) a) Find the domain of the transformed function. [select: ( or [, [, (] [input],[input][select: ) or ], ), ]] [math: (] \mbox{(} [math: −∞] -\infty ,[math: ∞] \infty [math: )] \mbox{)} b) Find the range of the transformed function. [select: ( or [, [, (] [input],[input][select: ) or ], ), ]] ,
Check Question 6

Question

BlackTom AI · Accepted Answer

The problem asks for two pieces of information about a transformed exponential function. To start, I restate what is given and what the transformation does:
- Original form (interpreted from the text): f(x) = 6^{6x + 4} + 1
- It is translated 5 units to the right along the x-axis. A horizontal translation does not change the y-values, only the input variable, so the function after translation is g(x) = 6^{6(x - 5) + 4} + 1 = 6^{6x - 30 + 4} + 1 = 6^{6x - 26} + 1.
- We are asked for (a) the domain and (b) the range of the transformed function g.

Domain analysis:
- Exponential expressions of the form a^{u(x)} are defined for all real x as long as the base a > 0 and a ≠ 1. Here the base is 6, which is positive and not equal to 1. The exponent is a real-valued linear expression in x, so there are no restrictions on x arising from the exponent.
- Translation left or right does not introduce any values of x that were previously invalid. Therefore, the domain remains all real numbers.
- In interval notation, this is (-∞, ∞).

Range analysis:
- The term 6^{6x - 26} is always positive for any real x, since an exponential with base 6 is never zero or negative.
- The smallest possible value of 6^{6x - 26} is approached as the exponent tends to -∞, in which case 6^{6x - 26} → 0^+. Consequently, g(x) = 6^{6x - 26} + 1 approaches 1 from above but never reaches 1.
- As x grows, the exponent 6x - 26 grows without bound, so 6^{6x - 26} becomes arbitrarily large, and thus g(x) → ∞.
- Therefore, the range is all real numbers greater than 1, not including 1 itself.
- In interval notation, the range is (1, ∞).

Why the given options are correct or incorrect:
- If a student considered domain as (-∞, ∞) for the transformed function, that is correct because the translation does not restrict x-values.
- If a student believed the domain would be something like [−∞, ∞) or included any finite boundaries, that would be mistaken unless explicitly restricted by the problem; here none exist.
- If someone claimed the range includes 1 or equals zero, that would be incorrect. The exponential term is strictly positive and the +1 shifts the entire range up, making the lower bound strictly greater than 1.
- If a student argued that the range starts at 0 or includes negative values, that would contradict the positive nature of 6^{6x - 26} and the +1 offset.

Summary check:
- Domain: all real numbers, (-∞, ∞).
- Range: (1, ∞).

MCD2130 - T2 - 2025 MCD2130 Test 3

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