In Exercises 51–66, find a. (fog) (x) b. (go f) (x) f(x) = 4-x, g(x) = 2x² +x+5

Verified step by step guidance

1. Step 1: Understand the composition of functions. The notation $(f\circ g)(x)$ means $f(g(x))$, and $(g\circ f)(x)$ means $g(f(x))$.
2. Step 2: For $(f\circ g)(x)$, substitute $g(x)$ into $f(x)$. Since $f(x)=4-x$ and $g(x)=2x^2+x+5$, replace $x$ in $f(x)$ with $g(x)$.
3. Step 3: Calculate $f(g(x))$ by substituting $g(x)=2x^2+x+5$ into $f(x)=4-x$, resulting in $f(g(x))=4-(2x^2+x+5)$.
4. Step 4: Simplify the expression $4-(2x^2+x+5)$ by distributing the negative sign and combining like terms.
5. Step 5: For $(g\circ f)(x)$, substitute $f(x)$ into $g(x)$. Replace $x$ in $g(x)$ with $f(x)=4-x$, resulting in $g(f(x))=2(4-x{)}^2+(4-x)+5$.

Key Concepts

Function Composition

Function composition involves combining two functions to create a new function. If f(x) and g(x) are two functions, the composition (fog)(x) means applying g first and then f to the result, expressed as f(g(x)). Conversely, (go f)(x) means applying f first and then g, written as g(f(x)). Understanding this concept is crucial for solving the given problem.

Evaluating Functions

Evaluating functions requires substituting a specific input value into the function's formula to find the output. For example, if f(x) = 4 - x, to evaluate f(2), you would substitute 2 for x, resulting in f(2) = 4 - 2 = 2. This skill is essential for calculating the results of the composed functions in the exercises.

Quadratic Functions

A quadratic function is a polynomial function of degree two, typically expressed in the form g(x) = ax² + bx + c. In this case, g(x) = 2x² + x + 5 is a quadratic function where a = 2, b = 1, and c = 5. Recognizing the properties of quadratic functions, such as their parabolic shape and vertex, is important for understanding their behavior when composed with other functions.