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.