Find left parenthesis lower f minus lower g right parenthesis left parenthesis x right parenthesis where lower f left parenthesis x right parenthesis equals 2 x squared, lower g left parenthesis x right parenthesis equals left parenthesis x plus 1 right parenthesis superscript 2.
To find the expression, we need to substitute the functions into the given equation.
Let's substitute:
- lower f(x) = 2x^2
- lower g(x) = (x+1)^2
So, the expression becomes:
left parenthesis lower f minus lower g right parenthesis left parenthesis x right parenthesis
Plugging in the values of lower f and lower g:
left parenthesis 2x^2 - (x+1)^2 right parenthesis
= 2x^2 - (x+1)(x+1)
= 2x^2 - (x^2 + 2x + 1)
= 2x^2 - x^2 - 2x - 1
= x^2 - 2x - 1
Therefore, the expression is:
x^2 - 2x - 1
Find lower f superscript negative 1 for the function lower f left parenthesis x right parenthesis equals left parenthesis x minus 5 right parenthesis superscript 3 baseline plus 8
To find the inverse of the function lower f, denoted as lower f^(-1), we need to follow these steps:
Step 1: Replace lower f(x) with y.
y = (x - 5)^3 + 8
Step 2: Swap the x and y variables.
x = (y - 5)^3 + 8
Step 3: Solve for y.
x - 8 = (y - 5)^3
(y - 5)^3 = x - 8
Step 4: Take the cube root of both sides.
y - 5 = ∛(x - 8)
Step 5: Solve for y.
y = ∛(x - 8) + 5
Therefore, the inverse function of lower f is lower f^(-1)(x) = ∛(x - 8) + 5.
To find \( f(x) - g(x) \), we need to evaluate \( f(x) \) and \( g(x) \) first.
Given:
\( f(x) = 2x^2 \)
\( g(x) = (x+1)^2 \)
To find the value of \( f(x) - g(x) \), we substitute the given functions into the equation:
\( f(x) - g(x) = 2x^2 - (x+1)^2 \)
Expanding the square:
\( f(x) - g(x) = 2x^2 - (x^2 + 2x + 1) \)
Now, simplify the expression:
\( f(x) - g(x) = 2x^2 - x^2 - 2x - 1 \)
Combining like terms:
\( f(x) - g(x) = x^2 - 2x - 1 \)
Therefore, \( f(x) - g(x) = x^2 - 2x - 1 \).