What is the domain of h(x) when h(x) = integral (-2, 5x+2) f(t)dt
and the domain of f(t) is [-2,3]
To find the domain of h(x), we need to consider the values of x that make the integral function meaningful.
The given integral is:
h(x) = ∫(from -2 to 5x+2) f(t) dt
The domain of f(t) is [-2, 3], which means that f(t) is defined for all values of t between -2 and 3 (inclusive).
To find the domain of h(x), we need to determine the values of x that will make the integral limits valid. In this case, the lower limit of integration is -2 and the upper limit of integration is 5x+2.
To ensure that the upper limit of integration, 5x+2, is within the valid domain of f(t), we check if it falls within the interval [-2, 3].
Let's solve the inequality:
-2 ≤ 5x+2 ≤ 3
Starting with the left inequality:
-2 ≤ 5x + 2
Subtracting 2 from both sides:
-4 ≤ 5x
Dividing by 5 (since 5 is positive, there is no need to change the direction of the inequality):
-4/5 ≤ x
Now, let's move to the right inequality:
5x + 2 ≤ 3
Subtracting 2 from both sides:
5x ≤ 1
Dividing by 5:
x ≤ 1/5
Therefore, the domain of h(x) is the intersection of the valid domains for f(t) and the values of x that satisfy the inequality:
Domain of h(x): -4/5 ≤ x ≤ 1/5