Let f be the function be defined by f(x)=5^((2x^2)-1)^(1/2)
A) is f an even or odd function? Justify answer.
B) find the domain of f.
C) find the range of f.
D) find f'(x)
To answer these questions, let's break them down step by step:
A) To determine whether a function is even or odd, we need to check if it satisfies the following properties:
1. Even function: f(x) = f(-x) (symmetric with respect to the y-axis)
2. Odd function: f(x) = -f(-x) (symmetric with respect to the origin)
In this case, let's substitute -x into the function and see if it satisfies either of the properties.
f(x) = 5^((2x^2)-1)^(1/2)
f(-x) = 5^((2(-x)^2)-1)^(1/2)
= 5^(2x^2 - 1)^(1/2)
Comparing f(x) and f(-x), we can see that they are not equal. Therefore, the function f(x) is neither even nor odd.
B) To find the domain of a function, we need to identify all the real numbers that x can take without causing any mathematical issues. For our function f(x), we need to consider any possible restrictions.
In this case, since we are dealing with exponentiation, the base must be positive (since 5^x is only defined for positive real numbers). Additionally, the argument inside the exponent (2x^2 - 1) must not result in a negative value under the square root.
To ensure these conditions are met, we have:
1. The base of the exponentiation: 5 > 0
2. The argument inside the exponent: 2x^2 - 1 ≥ 0
Solving the inequality: 2x^2 - 1 ≥ 0
2x^2 ≥ 1
x^2 ≥ 1/2
x ≥ sqrt(1/2) or x ≤ -sqrt(1/2)
Combining these conditions, the domain of f is given by:
Domain: (-∞, -sqrt(1/2)] ∪ [sqrt(1/2), +∞)
C) To find the range of a function, we need to identify all the possible values that the function can output. In this case, since we are dealing with exponentiation to the power of 1/2, the range will depend on the possible values of the base (5^((2x^2)-1)).
Since the base of the exponentiation is always positive, raising a positive base to any power will also result in a positive value. Therefore, the range of f will be all positive real numbers.
Range: (0, +∞)
D) To find the derivative of f(x), we need to apply the chain rule. The chain rule states that if we have a composite function, we need to differentiate the outer function first and then multiply it by the derivative of the inner function.
Given the function f(x) = 5^((2x^2)-1)^(1/2), let's find f'(x):
f(x) = 5^((2x^2)-1)^(1/2)
Applying the chain rule, let's differentiate the outer function first (5^u) and then multiply it by the derivative of the inner function ((2x^2) - 1):
f'(x) = (1/2) * ln(5) * 5^((2x^2)-1)^(1/2) * (2x^2 - 1)
Simplifying:
f'(x) = ln(5) * x^2 * 5^((2x^2)-1)^(1/2) - ln(5) * 5^((2x^2)-1)^(1/2)