Let z=[y] and y=[x]-x, where [.] is the greatest integer function. If x is <strong>NOT</strong> an integer but positive, then what is the value of z ?

Explanation

Since x is not an integer, we can write x = n + f where n is an integer and 0 \lt f \lt 1. Then [x] = n. Substituting this into y, we get y = n - (n + f) = -f. Because 0 \lt f \lt 1, we have -1 \lt -f \lt 0, meaning -1 \lt y \lt 0. The greatest integer less than or equal to y is -1. Thus, z = [y] = -1.

Related questions on Calculus

• If f(x)=4x+1 and g(x)=kx+2 such that fog(x)=gof(x), then what is the value of k?
• What is the <strong>MINIMUM</strong> value of the function f(x)=\log_{10}(x^{2}+2x+11)?
• What is \int(x^{x})^{2}(1+\ln x)\,dx equal to ?
• What is \int e^{x}\{1+\ln x+x\ln x\}\,dx equal to?
• What is \int\frac{(\cos x)^{1.5}-(\sin x)^{1.5}}{\sqrt{\sin x\cdot \cos x}}\,dx equal to ?