suppoe f is a function with domain [1,3]and range[2,5]?
define functions g and h by g(x)=3f(x)and h(x)=f(4x)
what is domain and range of g
what is domain and range of h
I solved it and got it like this
domainof g=[1,3]and range of g=[3,9]
Domain of h=[4,12] and range of h=[2,5]
Is it correct
domain of f is also the domain of g and h, since they are defined as multiples of f.
since range of f = [2,5],
range of g = [3*2,3*5] = [6,15]
range of h is [4*2,4*5] = [8,20]
range of g is 3*range of f, not 3*domain of f
domain is the same for all, since g and h are defined for exactly the same values as f is.
Yes, your solutions are correct.
The function g(x) is defined as g(x) = 3f(x). Since f(x) has a domain of [1,3], g(x) will also have the same domain of [1,3]. The function g(x) multiplies the output of f(x) by 3, so the range of g(x) will be three times the range of f(x), which is [2,5]. Therefore, the range of g(x) is [3,9].
The function h(x) is defined as h(x) = f(4x). Since f(x) has a domain of [1,3], h(x) is defined for all values of x that can be plugged into 4x and still give values within the domain of f(x). Solving the inequality 1 ≤ 4x ≤ 3, we get 1/4 ≤ x ≤ 3/4. Therefore, the domain of h(x) is [1/4, 3/4]. The function h(x) applies the input x to the function f(x) after multiplying it by 4, which means the range of h(x) will be the same as the range of f(x), which is [2,5]. Hence, the range of h(x) is [2,5].
Yes, your solutions are correct.
To determine the domain and range of a function g(x) = 3f(x), you need to consider the domain and range of the original function f(x).
Since f is a function with domain [1,3] and range [2,5], you can multiply the range values by 3 to find the range of g(x):
- The minimum value of the range of f, which is 2, multiplied by 3 gives the minimum value of the range of g, which is 6.
- The maximum value of the range of f, which is 5, multiplied by 3 gives the maximum value of the range of g, which is 15.
So, the range of g is [6,15].
For the domain of g(x), it remains the same as the domain of f(x), which is [1,3].
Now, let's consider the function h(x) = f(4x). In this case, the domain of h will be affected by the argument of f.
Notice that multiplying x by 4 in h(x) stretches the domain by a factor of 1/4. Thus, the domain of h(x) is [1/4, 3/4] when considering the original domain [1,3].
To determine the range of h(x), substitute the minimum and maximum values of the domain of h (1/4 and 3/4) into the function f(4x) and find the corresponding range. Since the range of f is [2,5], you need to evaluate f(4(1/4)) and f(4(3/4)).
f(4(1/4)) = f(1) = 2
f(4(3/4)) = f(3) = 5
Thus, the range of h is [2,5].
Therefore, your calculations:
- The domain of g is [1,3] and the range of g is [6,15].
- The domain of h is [1/4, 3/4] and the range of h is [2,5].