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

Verified step by step guidance

1. First, identify the given functions: \(f(x) = x^{2} - x + 4\) and \(g(x) = 3x - 5\).
2. Calculate \(g(1)\) by substituting \(x = 1\) into the function \(g(x)\): \(g(1) = 3(1) - 5\).
3. Simplify the expression for \(g(1)\) to find its value.
4. Next, find \(f(g(1))\) by substituting the value of \(g(1)\) into the function \(f(x)\): \(f(g(1)) = (g(1))^{2} - g(1) + 4\).
5. Simplify the expression for \(f(g(1))\) by squaring \(g(1)\), subtracting \(g(1)\), and then adding 4 to get the final expression.

Key Concepts

Function Evaluation

Function evaluation involves substituting a specific input value into the function's formula to find the corresponding output. For example, to find g(1), replace x with 1 in g(x) = 3x - 5 and simplify.

Composite Functions

A composite function like f(g(1)) means applying one function to the result of another. First, evaluate the inner function g(1), then use that output as the input for the outer function f.

Polynomial and Linear Functions

Understanding the types of functions involved helps in evaluation. Here, f(x) is a quadratic polynomial (degree 2), and g(x) is a linear function (degree 1), which affects how you substitute and simplify expressions.