Textbook Question
More composite functions Let ƒ(x) = | x | , g(x)= x² - 4 , F(x) = √x , G(x) = (1)/(x-2) Determine the following composite functions and give their domains.
G o g o ƒ
0. Functions
Combining Functions
Problem 60
Textbook Question
Missing piece Let g(x) = x² + 3 Find a function ƒ that produces the given composition.
(g o ƒ ) (x) = x²⸍³ + 3
Verified step by step guidance1
Step 1: Understand the composition (g \circ f)(x) = g(f(x)). We need to find a function f(x) such that when g is applied to f(x), it results in x^{2/3} + 3.
Step 2: Recall that g(x) = x^2 + 3. We want g(f(x)) = f(x)^2 + 3 to equal x^{2/3} + 3.
Step 3: Set up the equation f(x)^2 + 3 = x^{2/3} + 3.
Step 4: Subtract 3 from both sides to isolate the squared term: f(x)^2 = x^{2/3}.
Step 5: Solve for f(x) by taking the square root of both sides: f(x) = \sqrt{x^{2/3}}.
Was this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Function Composition
Function composition involves combining two functions where the output of one function becomes the input of another. In this case, the composition (g o ƒ)(x) means applying function ƒ first and then applying function g to the result. Understanding how to manipulate and combine functions is essential for solving problems involving compositions.
Recommended video:
3:48
Evaluate Composite Functions - Special Cases
Identifying Functions
To find the function ƒ that satisfies the composition (g o ƒ)(x) = x² + 3, we need to identify the structure of g(x) and how it relates to ƒ. Here, g(x) = x² + 3 suggests that ƒ must produce an input that, when squared and increased by 3, results in the desired output. Recognizing the form of g helps in determining the appropriate form of ƒ.
Recommended video:
6:13Exponential Functions
Inverse Functions
Inverse functions are crucial in understanding how to 'reverse' the operations of a function. If we can express g(x) in terms of its inverse, we can derive ƒ by manipulating the equation. For instance, if we can isolate x in terms of g, we can find the function that, when composed with g, yields the original input, aiding in solving the composition problem.
Recommended video:
4:49Inverse Cosine
5:56mWatch next
Master Adding & Subtracting Functions with a bite sized video explanation from Patrick
Start learningRelated Videos
Related Practice