In Exercises 95–96, find all values of x satisfying the given conditions. f(x) = 2x − 5, g(x) = x² − 3x + 8, and (ƒ o g) (x) = 7.

Verified step by step guidance

1. First, understand that \((f \circ g)(x)\) means \(f(g(x))\). This means we need to substitute \(g(x)\) into \(f(x)\).
2. Given \(f(x) = 2x - 5\) and \(g(x) = x^2 - 3x + 8\), substitute \(g(x)\) into \(f(x)\) to get \(f(g(x)) = 2(g(x)) - 5\).
3. Substitute \(g(x) = x^2 - 3x + 8\) into \(f(x)\): \(f(g(x)) = 2(x^2 - 3x + 8) - 5\).
4. Simplify the expression: \(2(x^2 - 3x + 8) - 5 = 2x^2 - 6x + 16 - 5\).
5. Set the expression equal to 7, as given: \(2x^2 - 6x + 16 - 5 = 7\), and solve for \(x\).

Key Concepts

Function Composition

Function composition involves combining two functions to create a new function. The notation (ƒ o g)(x) means to apply g first and then apply f to the result of g. This is essential for solving the problem, as it requires evaluating f at the output of g, which is a fundamental operation in algebra.

Solving Equations

Solving equations is the process of finding the values of variables that satisfy a given mathematical statement. In this context, we need to set the composed function (ƒ o g)(x) equal to 7 and solve for x. This involves manipulating the equation to isolate x, which is a critical skill in algebra.

Quadratic Functions

Quadratic functions are polynomial functions of degree two, typically expressed in the form g(x) = ax² + bx + c. In this problem, g(x) is a quadratic function, and understanding its properties, such as its vertex and roots, is important for analyzing the output of g and how it interacts with f in the composition.