The domain of a function f(x) is {x∈R|−4≤x≤16}. The range of f(x) is {y∈R| −8≤y≤12}
Given g(x) = 3/4f(−x+3) + 5, what is the domain and range of g(x) ?
please help show with steps!
To find the domain and range of g(x), we need to consider the restrictions imposed by the function g(x) as well as the domain and range of f(x).
1. Start with the domain of f(x): {x∈R|−4≤x≤16}. Since g(x) is based on f(−x+3), we need to find the values of x for which −x+3 is within the domain of f(x).
- Solve the inequality −4≤−x+3≤16:
- Subtract 3 from all sides: −7≤−x≤13
- Multiply by −1 to change the direction of the inequalities: 7≥x≥−13
- Therefore, the domain of g(x) is {x∈R|−13≤x≤7}.
2. Next, consider the range of f(x): {y∈R|−8≤y≤12}. To find the range of g(x), we need to determine the possible range of values when we apply the given function expression to the range of f(x). Recall that the expression is 3/4f(−x+3) + 5.
- Substitute the minimum value of f(x) into the expression: 3/4(-8) + 5 = -6 + 5 = -1
- Substitute the maximum value of f(x) into the expression: 3/4(12) + 5 = 9 + 5 = 14
- Therefore, the range of g(x) is {y∈R|−1≤y≤14}.
In summary:
- The domain of g(x) is {x∈R|−13≤x≤7}.
- The range of g(x) is {y∈R|−1≤y≤14}.