True or False:
1. f(x) = -x^3 + x + 1 is an odd function
2. f(x) = x^3 -ax^2 + bx represents a cubic equation having no constant term ...(I'm thinking true?)
3. If f(x) = x/x+2 then f^-1 (x) = 2x/1-x ...(my guess is false?)
4. The equation x^3 - 12x + 16 = 0 has a double root at x = -4
1. check if f(x)=-f(-x)
2. correct
3. If f(x)=x/(x+2)
solve for y in x=y/(y+2) where y=f-1(x)
4. Substitute x=-4 in the equation and see if the left-hand-side vanishes.
true
Let's go through each statement one by one and explain how to determine the answers for each:
1. To determine whether a function is odd or not, we need to check if f(-x) = -f(x) for every value of x in the function's domain. For the given function f(x) = -x^3 + x + 1, let's substitute -x for x and see if the equation holds true:
f(-x) = -(-x)^3 + (-x) + 1
= -(-x^3) - x + 1
= -(-x^3) + (-x) + 1
= x^3 - x + 1
Since f(-x) is not equal to -f(x) (which is -(-x^3) + x - 1), we can conclude that the given function is not odd. Therefore, the statement is false.
2. To determine whether a cubic equation has a constant term or not, we need to look at the equation in the form f(x) = ax^3 + bx^2 + cx + d, where d represents the constant term. In the given equation f(x) = x^3 - ax^2 + bx, there is no explicit constant term (d = 0). Therefore, the statement is true.
3. To find the inverse of a function, let's start with the given function f(x) = x/(x+2). To determine if the statement is true or false, let's try to find the inverse function.
Step 1: Replace f(x) with y.
y = x/(x+2)
Step 2: Swap x and y.
x = y/(y+2)
Step 3: Solve for y.
x(y+2) = y
xy + 2x = y
xy - y = -2x
y(x - 1) = -2x
y = (-2x)/(x - 1) or y = -2x/(x - 1)
So, the inverse function of f(x) = x/(x+2) is f^(-1)(x) = -2x/(x - 1). Therefore, the statement is true.
4. To determine whether the given equation x^3 - 12x + 16 = 0 has a double root at x = -4, we need to find the derivative of the equation and check if it has a root at x = -4.
Step 1: Find the derivative of the equation with respect to x:
d/dx (x^3 - 12x + 16) = 3x^2 - 12
Step 2: Evaluate the derivative at x = -4:
3(-4)^2 - 12 = 48 - 12 = 36
Since the derivative evaluated at x = -4 is not zero, we can conclude that the equation does not have a double root at x = -4. Therefore, the statement is false.