Let f be the function be defined by f(x)=5^((2x^2)-1)^(1/2)
find the range of f without using a calculator
f(x)=5^((2x^2)-1)^(1/2)
= 5^(x^2 - 1/2)
x^2 is always positive, so the smallest exponent that 5 could have is -1/2 when x = 0
for any other value of x, x^2 - 1/2 is greater than -1/2
we know 5^(-1/2) = 1/√5 or √5/5
so the range is y ≥ √5/5
To find the range of the function f(x) without using a calculator, we need to consider the properties and behavior of the function.
First, notice that the function f(x) involves a composition of several algebraic operations, such as exponentiation and square root. We can break down the function into its different parts to analyze it more easily.
Let's consider the innermost expression: (2x^2) - 1. This expression is a quadratic function of x, which means that it will always be non-negative. In other words, (2x^2) - 1 ≥ 0 for all real values of x.
Taking the next operation, 5 raised to the power of ((2x^2) - 1), we know that 5^y is an increasing function for y > 0. Thus, as ((2x^2) - 1) is non-negative (∵ (2x^2) - 1 ≥ 0), the expression 5^((2x^2) - 1) is also non-negative.
Finally, we have the outer operation, which is taking the square root of 5^((2x^2) - 1). We know that the square root function returns non-negative values. Hence, the range of f(x) is also non-negative.
Therefore, the range of f(x) is all real numbers greater than or equal to 0, i.e., [0, +∞).