Let f(x) = x² − x + 4 and g(x) = 3x – 5. Find g(-1) and f(g(-1)).

Verified step by step guidance

1. Step 1: Start by evaluating g(-1). Substitute x = -1 into the function g(x) = 3x - 5. This means you will calculate g(-1) = 3(-1) - 5.
2. Step 2: Simplify the expression for g(-1). Multiply 3 by -1 and then subtract 5. This will give you the value of g(-1).
3. Step 3: Use the result from Step 2 as the input for the function f(x). Substitute g(-1) into f(x) = x² - x + 4. This means you will calculate f(g(-1)) = (g(-1))² - g(-1) + 4.
4. Step 4: Expand and simplify the expression for f(g(-1)). Square the value of g(-1), subtract g(-1), and then add 4.
5. Step 5: Combine all terms to simplify the expression for f(g(-1)). This will give you the final value of f(g(-1)).

Key Concepts

Function Evaluation

Function evaluation involves substituting a specific input value into a function to determine its output. For example, to find g(-1), we replace x in the function g(x) = 3x - 5 with -1, resulting in g(-1) = 3(-1) - 5 = -8. Understanding how to evaluate functions is crucial for solving problems that require finding specific outputs based on given inputs.

Composition of Functions

The composition of functions occurs when the output of one function becomes the input of another. In this case, after finding g(-1), we use that result as the input for the function f. The notation f(g(-1)) indicates that we first evaluate g at -1 and then substitute that output into f, which is essential for solving problems that involve multiple functions.

Quadratic Functions

A quadratic function is a polynomial function of degree two, typically expressed in the form f(x) = ax² + bx + c. In this problem, f(x) = x² - x + 4 is a quadratic function where a = 1, b = -1, and c = 4. Understanding the properties of quadratic functions, such as their parabolas and vertex, is important for analyzing their behavior and outputs.