Question

Determine the transformations that produce the graph of the functions g(x)=0.2log(x+14)+10 and h(x)=5log(x+14)−10 from the parent function f(x)=logx. Then compare the similarities and differences between the two functions, including the domain and range.

Answer 1

Let's focus on g(x) for now.

0.2 = 1/5

f(x) = log(x) is the parent function

0.2*f(x) = 0.2*log(x) .... vertical compression by factor of 5

0.2*f(x+14) = 0.2*log(x+14) .... shift 14 units to the left

0.2*f(x+14)+10 = 0.2*log(x+14)+10 .... shift 10 units up

g(x) = 0.2*log(x+14)+10

The transformations to go from f(x) to g(x) are:

1. vertical compression by factor of 5
2. shift 14 units to the left
3. shift 10 units up

The domain of g(x) is x > -14 to ensure that x+14 > 0. This is so we avoid applying a log to 0 or a negative value. The parent function domain was x > 0. Note how the domain shifted 14 units to the left (step 2)

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Now onto h(x)

f(x) = log(x)

5f(x) = 5log(x) .... vertical stretch by factor of 5

5f(x+14) = 5log(x+14) ... shift 14 units left, just like earlier

5f(x+14)-10 = 5log(x+14)-10 ... shift 10 units down

h(x) = 5log(x+14)-10

The transformations to go from f(x) to h(x) are:

1. vertical stretch by factor of 5
2. shift 14 units to the left (same as g(x))
3. shift 10 units down (instead of up compared to g(x))

This time h(x) is taller compared to f(x), while g(x) is more vertically squished compared to f(x). Both involve being shifted 14 units to the left, so both g(x) and h(x) have the same domain. Vertical shifting and compressing/stretching does not affect the domain.

The range of f(x), g(x) and h(x) is the set of all real numbers.