Let f: R^+ →R be defined as f(x)=x^2-x+2 and g:[1,2] →[1,2] be defined as g(x)={x}+1, where {,}=

step 1 Hi guys, so we have this problem in which we need to find out basically F of GX and then there a

Let f: R^+ →R be defined as f(x)=x^2-x+2 and g:[1,2] →[1,2]...

Let $f: R^{+} \rightarrow R$ be defined as $f(x)=x^{2}-x+2$ and $g:[1,2] \rightarrow[1,2]$ be defined as $g(x)=\{x\}+1$, where $\{,\}=$, Fractional part function. If the domain and range of $f(g(x))$ are $[a, b]$ and $[c, d)$, then find the value of $\frac{b}{a}+\frac{d}{c}$

Key Concepts

Function Composition

Function composition involves applying one function to the results of another. In this context, f(g(x)) means that for any input x from the domain of g, the output of g is used as the input for f. This technique is fundamental in understanding how changes in one function affect the overall mapping when functions are combined.

Domain and Range Determination

Determining the domain and range of a composite function requires identifying the set of all valid inputs for the inner function and then tracking how these inputs are transformed by the outer function. This process involves understanding the restrictions and behavior of both functions involved to ascertain the final set of inputs (domain) and outputs (range) of the composite mapping.

Fractional Part Function

The fractional part function, denoted by {x}, extracts the decimal part of a real number, discarding its integer part. It plays an important role in problems by introducing periodic or piecewise behavior, and understanding its properties is critical when it appears as part of a more complex function.

Quadratic Function Analysis

Quadratic functions, which generally take the form ax² + bx + c, have key properties such as a vertex and a parabolic shape that can either open upwards or downwards depending on the coefficient a. Analyzing a quadratic function over a specific interval is important for determining its minimum and maximum values, which in turn is essential in finding the range of a composed function.

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