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.
To determine the transformations that produce the graphs of g(x) and h(x) from the parent function f(x)=log(x), we need to compare the equations of g(x) and h(x) with the parent function.
For g(x), we have g(x) = 0.2log(x+14) + 10.
The transformation that affects the function f(x)=log(x) is the horizontal shift. In this case, the graph of g(x) is shifted left by 14 units. This is because the argument of the logarithm, x+14, is replacing x in f(x).
For h(x), we have h(x) = 5log(x+14) - 10.
Similar to g(x), h(x) also involves a horizontal shift left by 14 units. Additionally, the graph of h(x) also undergoes a vertical shift down by 10 units. This is because the entire function h(x) is multiplied by 5 and then subtracted by 10.
Comparing the similarities and differences between g(x) and h(x):
Similarities:
- Both functions have a horizontal shift left by 14 units.
- Both functions involve a logarithmic transformation.
Differences:
- The function g(x) has a vertical shift up by 10 units, while h(x) has a vertical shift down by 10 units.
- The function g(x) has a multiplier of 0.2 for the logarithmic part, while h(x) has a multiplier of 5.
Domain and Range:
The domain for both g(x) and h(x) is restricted by the argument of the logarithm. Since x+14 must be greater than 0 (to avoid taking the logarithm of a negative number), the domain for both functions is x > -14.
The range for g(x) is given by the vertical shift up by 10 units. The range for h(x) is given by the vertical shift down by 10 units. Thus, the range for g(x) is y > 10 and the range for h(x) is y < -10.