Find the Inverse f(x) = square root of x+2+9
The question is asking you to determine the inverse function of the given function f(x), which is defined as the square root of (x+2), and then increased by 9. In other words, you need to find a function that 'undoes' what f(x) does, so that if you apply this new function to the result of f(x), you would get the original input value x. The process typically involves swapping x and f(x), solving for the new x, and then properly defining the resulting expression as the inverse function f^(-1)(x).
$f \left(\right. x \left.\right) = \sqrt{x + 2 + 9}$
Answer
Solution:
Step 1:
Express $f(x) = \sqrt{x + 2 + 9}$ as an equation: $y = \sqrt{x + 11}$.
Step 2:
Swap the roles of $x$ and $y$: $x = \sqrt{y + 11}$.
Step 3:
Isolate $y$.
Step 3.1:
Rewrite the equation: $x = \sqrt{y + 11}$.
Step 3.2:
Eliminate the square root by squaring both sides: $(\sqrt{y + 11})^2 = x^2$.
Step 3.3:
Simplify the equation.
Step 3.3.1:
Express $\sqrt{y + 11}$ as $(y + 11)^{\frac{1}{2}}$: $((y + 11)^{\frac{1}{2}})^2 = x^2$.
Step 3.3.2:
Simplify the left-hand side.
Step 3.3.2.1:
Apply the exponent rule: $((y + 11)^{\frac{1}{2}})^2$.
Step 3.3.2.1.1:
Multiply the exponents: $(y + 11)^{\frac{1}{2} \cdot 2} = x^2$.
Step 3.3.2.1.1.1:
Use the power of a power rule: $(y + 11)^{1} = x^2$.
Step 3.3.2.1.2:
Combine terms: $y + 11 = x^2$.
Step 3.4:
Subtract 11 from both sides: $y = x^2 - 11$.
Step 4:
Replace $y$ with $f^{-1}(x)$: $f^{-1}(x) = x^2 - 11$.
Step 5:
Confirm that $f^{-1}(x) = x^2 - 11$ is indeed the inverse of $f(x) = \sqrt{x + 11}$.
Step 5.1:
Check if $f^{-1}(f(x)) = x$ and $f(f^{-1}(x)) = x$.
Step 5.2:
Compute $f^{-1}(f(x))$.
Step 5.2.1:
Set up the composite function: $f^{-1}(f(x))$.
Step 5.2.2:
Substitute $f(x)$ into $f^{-1}(x)$: $f^{-1}(\sqrt{x + 11}) = (\sqrt{x + 11})^2 - 11$.
Step 5.2.3:
Simplify the equation: $f^{-1}(\sqrt{x + 11}) = x + 11 - 11$.
Step 5.2.4:
Combine like terms: $f^{-1}(\sqrt{x + 11}) = x$.
Step 5.3:
Compute $f(f^{-1}(x))$.
Step 5.3.1:
Set up the composite function: $f(f^{-1}(x))$.
Step 5.3.2:
Substitute $f^{-1}(x)$ into $f(x)$: $f(x^2 - 11) = \sqrt{(x^2 - 11) + 11}$.
Step 5.3.3:
Simplify the equation: $f(x^2 - 11) = \sqrt{x^2}$.
Step 5.3.4:
Assuming $x$ is non-negative, simplify the square root: $f(x^2 - 11) = x$.
Step 5.4:
Since both $f^{-1}(f(x)) = x$ and $f(f^{-1}(x)) = x$, we confirm that $f^{-1}(x) = x^2 - 11$ is the correct inverse of $f(x) = \sqrt{x + 11}$.
Knowledge Notes:
Inverse Functions: The inverse of a function $f(x)$, denoted as $f^{-1}(x)$, is a function that reverses the effect of $f(x)$. If $f(a) = b$, then $f^{-1}(b) = a$.
Interchanging Variables: To find the inverse function, we interchange the dependent variable ($y$) with the independent variable ($x$).
Solving for the Inverse: To solve for the inverse, we manipulate the equation to isolate the new dependent variable (formerly the independent variable).
Squaring Both Sides: When a variable is under a square root, we square both sides of the equation to eliminate the square root.
Exponent Rules: The power of a power rule states that $(a^m)^n = a^{m \cdot n}$. When we have an exponent of 1, it means the expression remains the same, as $a^1 = a$.
Verification: To verify that two functions are inverses, we check if $f^{-1}(f(x)) = x$ and $f(f^{-1}(x)) = x$. If both conditions are satisfied, the functions are indeed inverses of each other.
