Problem 87
Question
In Exercises \(87-96,\) find two functions \(f\) and \(g\) such that \(h(x)=(f \circ g)(x)=f(g(x)) .\) Answers may vary. $$h(x)=(3 x-1)^{2}$$
Step-by-Step Solution
Verified Answer
To find two functions \(f\) and \(g\) such that \(h(x) = f (g(x))\) for \(h(x) = (3x -1)^2\), we can say that \(f(x) = x^{2}\) and \(g(x) = 3x-1\).
1Step 1: Identifying Function \(g\)
Observe inside the brackets of \(h(x)\). You see a linear function represented \(g(x)\) which is \(3x-1\). So, let's say \(g(x) = 3x - 1\).
2Step 2: Identifying Function \(f\)
Now look at the operation applied to the output of \(g(x)\) in \(h(x)\). The operation that is applied to \(g(x)\) is squaring, which will be our function \(f(x)\). Therefore, let's say \(f(x) = x^{2}\).
3Step 3: Function Composition
To confirm the functions \(f\) and \(g\) are correct, we perform a function composition operation which is \(f(g(x)) = ((3x - 1)^2) = (f \circ g)(x)\). This should be equal to the given \(h(x)\).
Key Concepts
Linear FunctionsQuadratic FunctionsNested Functions
Linear Functions
Linear functions are the simplest type of functions. They are equations of the form \(f(x) = ax + b\), where \(a\) and \(b\) are constants. These functions graph as straight lines and are the foundation for understanding more complex functions.
In the exercise, the linear function is identified as \(g(x) = 3x - 1\). Here, the slope \(a\) is 3 and the y-intercept \(b\) is -1. This tells us that for every unit increase in \(x\), the value of \(g(x)\) increases by 3. This straight line passes through the point where the value of \(y\) is -1 when \(x\) is 0. Linear functions are crucial in breaking down complex functions into easier steps, allowing us to manipulate and analyze functions part by part.
In the exercise, the linear function is identified as \(g(x) = 3x - 1\). Here, the slope \(a\) is 3 and the y-intercept \(b\) is -1. This tells us that for every unit increase in \(x\), the value of \(g(x)\) increases by 3. This straight line passes through the point where the value of \(y\) is -1 when \(x\) is 0. Linear functions are crucial in breaking down complex functions into easier steps, allowing us to manipulate and analyze functions part by part.
Quadratic Functions
Quadratic functions take the form \(f(x) = ax^2 + bx + c\). They graph as parabolas and can open upwards or downwards depending on the sign of \(a\). Quadratic functions are common in physics and engineering as they model phenomena like projectile motion.
In our exercise's context, the quadratic focus is on the squaring operation \(f(x) = x^2\). This specific quadratic function operates by taking any input value \(x\) and squaring it. Thus, its graph is a parabola that opens upwards because \(a=1\), and it is the simplest form of a quadratic equation, having no linear \(bx\) or constant \(c\) terms. Quadratic functions help form the basis for understanding how functions react to being raised to powers.
In our exercise's context, the quadratic focus is on the squaring operation \(f(x) = x^2\). This specific quadratic function operates by taking any input value \(x\) and squaring it. Thus, its graph is a parabola that opens upwards because \(a=1\), and it is the simplest form of a quadratic equation, having no linear \(bx\) or constant \(c\) terms. Quadratic functions help form the basis for understanding how functions react to being raised to powers.
Nested Functions
Nested functions involve taking one function and placing it inside another function, such as \(f(g(x))\). This is also known as function composition, a powerful tool in mathematics for creating complex operations from simpler ones.
In our example, the nested function is \(h(x) = (3x - 1)^2\) which can be broken down into a linear function \(g(x)\) nested inside a quadratic function \(f(x)\). The composition \((f \circ g)(x)\) demonstrates how these functions work together: first, the input \(x\) is transformed by \(g(x)\), creating \(3x - 1\), and then \(f(x)\) applies the squaring operation to this result. This layered approach helps solve complex problems by simplifying operations step by step, hence making it easily understandable and more manageable.
In our example, the nested function is \(h(x) = (3x - 1)^2\) which can be broken down into a linear function \(g(x)\) nested inside a quadratic function \(f(x)\). The composition \((f \circ g)(x)\) demonstrates how these functions work together: first, the input \(x\) is transformed by \(g(x)\), creating \(3x - 1\), and then \(f(x)\) applies the squaring operation to this result. This layered approach helps solve complex problems by simplifying operations step by step, hence making it easily understandable and more manageable.
Other exercises in this chapter
Problem 86
In Exercises \(67-86,\) find expressions for \((f \circ g)(x)\) and \((g \circ f)(x)\) Give the domains of \(f \circ g\) and \(g \circ f\). $$f(x)=\frac{-x+1}{2
View solution Problem 86
Solve the quadratic equation by entering the quadratic formula in the home screen of your graphing utility. (See Technology Note on page 167.) $$0.62 t^{2}-1.29
View solution Problem 88
In Exercises \(87-96,\) find two functions \(f\) and \(g\) such that \(h(x)=(f \circ g)(x)=f(g(x)) .\) Answers may vary. $$h(x)=(-2 x+5)^{2}$$
View solution Problem 89
In Exercises \(87-96,\) find two functions \(f\) and \(g\) such that \(h(x)=(f \circ g)(x)=f(g(x)) .\) Answers may vary. $$h(x)=\sqrt[3]{4 x^{2}-1}$$
View solution