Let f: R →R, g: R →R be two functions given by f(x)=2 x-3, g(x)=x^3+5. Then (f o g)^-1(x) is equ

step 1 so in this question we have in this question we have that y is equal to f of g of x that will be

Let f: R →R, g: R →R be two functions given by f(x)=2 x-3,...

Let $f: R \rightarrow R, g: R \rightarrow R$ be two functions given by $f(x)=2 x-3, g(x)=x^{3}+5$. Then $(f o g)^{-1}(x)$ is equal to (A) $\left(\frac{x-7}{2}\right)^{1}$ (B) $\left(\frac{x+7}{2}\right)^{1}$ (C) $\left(x-\frac{7}{2}\right)^{1 / 3}$ (D) $\left(\frac{x-2}{7}\right)^{1 / 3}$

Let $f: R \rightarrow R, g: R \rightarrow R$ be two functions given by $f(x)=2 x-3, g(x)=x^{3}+5$. Then $(f o g)^{-1}(x)$ is equal to
(A) $\left(\frac{x-7}{2}\right)^{1}$
(B) $\left(\frac{x+7}{2}\right)^{1}$
(C) $\left(x-\frac{7}{2}\right)^{1 / 3}$
(D) $\left(\frac{x-2}{7}\right)^{1 / 3}$

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Function Composition

This concept involves applying one function to the result of another. In the context of h = f o g, the output of g(x) becomes the input for f. Understanding composition is crucial as it allows the transformation of operations into a single function whose inverse may be derived systematically.

Inverse Function

An inverse function reverses the effect of the original function, meaning that if h(x) = (f o g)(x), then h?¹(x) will recover the input from the output. Finding the inverse requires setting h(x) equal to a variable and solving for the original input variable.

Algebraic Manipulation

Determining the inverse of a composed function typically involves standard algebraic techniques such as isolating terms, undoing operations (like addition, multiplication), and applying the appropriate inverse operations (such as taking cube roots when a cubic term is present).

Cubic Functions and Cube Roots

When the function in question includes a cubic term, its inversion necessitates the use of cube roots. Understanding how to solve equations involving cubic expressions and extract cube roots is key to obtaining the correct inverse function.

Transcript

Please give Ace some feedback