Find f(h(x)) f(x)=13-x h(x)=x^2-6x-18

Solution:

Step 1:

Construct the composite function $f(h(x))$.

Step 2:

Insert $h(x)=x^2-6x-18$ into $f(x)$ to get $f(h(x))$.

$$f(h(x))=f(x^2-6x-18)=13-(x^2-6x-18)$$

Step 3:

Begin simplifying the expression.

Step 3.1:

Utilize the distributive property to expand the terms.

$$f(h(x))=13-x^2+6x+18$$

Step 3.2:

Proceed with the simplification.

Step 3.2.1:

Combine like terms by adding $-6x$ to $6x$.

$$f(h(x))=13-x^2+6x+18$$

Step 3.2.2:

Combine like terms by adding $-18$ to $18$.

$$f(h(x))=13-x^2+6x+18$$

Step 4:

Combine the constants $13$ and $18$.

$$f(h(x))=-x^2+6x+31$$

Knowledge Notes:

The process outlined above involves creating a composite function, which is a function that is formed by substituting one function into another. In this case, we are substituting $h(x)$ into $f(x)$ to create $f(h(x))$. The steps involve:

1. Setting up the composite function: This step involves understanding the concept of function composition, where you replace the input of one function with another function.

2. Substitution: This is where you replace the variable in the outer function with the entire inner function. It is important to maintain the integrity of the inner function by using parentheses to ensure proper substitution.

3. Simplification: This step involves several sub-steps:

   • Applying the distributive property, which states that $a(b+c)=ab+ac$.

   • Simplifying the expression by combining like terms and performing arithmetic operations.

4. Final expression: The last step is to write the simplified form of the composite function, which is the result of the problem-solving process.

Understanding function composition is essential in various fields of mathematics, including algebra, calculus, and functional analysis. It allows us to create more complex functions and understand how different functions can interact with each other.