Find the Inverse f(x) = square root of x+2+9

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:

1. 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 $f^{-1}(b)=a$.

2. Interchanging Variables: To find the inverse function, we interchange the dependent variable ($y$) with the independent variable ($x$).

3. Solving for the Inverse: To solve for the inverse, we manipulate the equation to isolate the new dependent variable (formerly the independent variable).

4. Squaring Both Sides: When a variable is under a square root, we square both sides of the equation to eliminate the square root.

5. 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 $a^1=a$.

6. 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.