The functions f and g are defined as follows. =fx+5×2+4×3 and =gx−x25x Find f1x and g−x2 . Write your answers without parentheses and simplify them as much as possible.
The Correct Answer and Explanation is:
Let’s first look at the given functions:
- f(x)=x+5×2+4x3f(x) = x + 5x^2 + 4x^3f(x)=x+5×2+4×3
- g(x)=x−x25g(x) = x – \frac{x^2}{5}g(x)=x−5×2
Now, we need to find:
- f(1x)f(1x)f(1x) — This represents substituting xxx with 1x1x1x, or just xxx, in the function f(x)f(x)f(x). f(1x)=f(x)=x+5×2+4x3f(1x) = f(x) = x + 5x^2 + 4x^3f(1x)=f(x)=x+5×2+4×3 Since f(x)f(x)f(x) is already expressed in terms of xxx, the result for f(1x)f(1x)f(1x) is simply: f(1x)=x+5×2+4x3f(1x) = x + 5x^2 + 4x^3f(1x)=x+5×2+4×3
- g(−x2)g(-x^2)g(−x2) — This represents substituting xxx with −x2-x^2−x2 in the function g(x)g(x)g(x). g(−x2)=−x2−(−x2)25g(-x^2) = -x^2 – \frac{(-x^2)^2}{5}g(−x2)=−x2−5(−x2)2 Simplifying the expression: g(−x2)=−x2−x45g(-x^2) = -x^2 – \frac{x^4}{5}g(−x2)=−x2−5×4
So, the final answers are:
- f(1x)=x+5×2+4x3f(1x) = x + 5x^2 + 4x^3f(1x)=x+5×2+4×3
- g(−x2)=−x2−x45g(-x^2) = -x^2 – \frac{x^4}{5}g(−x2)=−x2−5×4
Explanation:
For f(1x)f(1x)f(1x), since there is no transformation needed other than replacing xxx with 1x1x1x, the function stays the same. The expression f(x)=x+5×2+4x3f(x) = x + 5x^2 + 4x^3f(x)=x+5×2+4×3 remains unchanged when we evaluate f(1x)f(1x)f(1x).
For g(−x2)g(-x^2)g(−x2), we substitute −x2-x^2−x2 into the expression for g(x)g(x)g(x). After substituting, we have to simplify the terms. The square of −x2-x^2−x2 is just x4x^4×4, so the final result is −x2−x45-x^2 – \frac{x^4}{5}−x2−5×4.
