Problem 9 - Applied Calculus

Question

For the functions \(f\) and \(g\) in Problems \(7-10,\) find (a) \(\quad f(g(1))\) (b) \(g(f(1))\) (c) \(\quad f(g(x))\) (d) \(g(f(x))\) (e) \(f(t) g(t)\) $$f(x)=e^{x}, g(x)=x^{2}$$

Step-by-Step Solution

Answer

(a) \(e\); (b) \(e^2\); (c) \(e^{x^2}\); (d) \(e^{2x}\); (e) \(e^t t^2\)."

Step 1: Evaluate f(g(1))

To find \(f(g(1))\), we first need to evaluate \(g(1)\). Since \(g(x) = x^2\), we have \(g(1) = 1^2 = 1\). Now substitute \(g(1) = 1\) into \(f(x) = e^x\): \(f(g(1)) = f(1) = e^1 = e\).

Step 2: Evaluate g(f(1))

To find \(g(f(1))\), we first evaluate \(f(1)\). Given \(f(x) = e^x\), we have \(f(1) = e^1 = e\). Now substitute \(f(1) = e\) into \(g(x) = x^2\): \(g(f(1)) = g(e) = e^2\).

Step 3: Find the expression f(g(x))

For \(f(g(x))\), we need to substitute \(g(x) = x^2\) into \(f(x) = e^x\). So, \(f(g(x)) = f(x^2) = e^{x^2}\).

Step 4: Find the expression g(f(x))

For \(g(f(x))\), substitute \(f(x) = e^x\) into \(g(x) = x^2\). So, \(g(f(x)) = g(e^x) = (e^x)^2 = e^{2x}\).

Step 5: Evaluate f(t) g(t)

To find \(f(t)g(t)\), multiply \(f(t) = e^t\) by \(g(t) = t^2\). So, \(f(t)g(t) = e^t \cdot t^2\).

Key Concepts

Functions Evaluation

When you deal with functions, evaluating them means finding out what value they give you when you input a specific number. It is like asking, "What will the output be if I plug in this number?" For example, if you have a function like

• \( f(x) = e^x \)

This is an exponential function. You might be asked to evaluate \( f(1) \), which means substituting 1 into the function. So, you'd calculate \( f(1) = e^1 = e \).Turning to our problem, we also have another function:

• \( g(x) = x^2 \)

This is a quadratic function, which we'll cover more later. If you want to evaluate \( g(1) \), you'd substitute 1 into this function to get \( g(1) = 1^2 = 1 \). This step-by-step approach helps you break down complex problems into manageable bits.

Exponential Function

An exponential function is one where a constant base is raised to a variable exponent. These functions grow rapidly, making them quite important in various fields like biology, finance, and physics. In our exercise,

• \( f(x) = e^x \)

is the exponential function where "\( e \)" is the base, known as Euler's number. It is approximately 2.71828. To understand how it works, think of \( e^x \) as repeated multiplication of \( e \) by itself \( x \) times. If \( x = 2 \), then \( e^2 = e \times e \). This rapid growth is why exponential functions are useful for modeling situations where growth accelerates, like population growth or interest compounding.In the exercise, you saw \( f(g(x)) \) being calculated. By plugging \( g(x) \) into \( f(x) \), you got an exponential term \( e^{x^2} \), adding another layer of complexity and growth.

Quadratic Function

Quadratic functions are polynomial functions of degree 2. They commonly appear in forms like

• \( g(x) = x^2 \)

The graph of a quadratic function is a parabola, which can open upwards or downwards.In the context of our exercise, it plays a pivotal role by being a building block for composite functions. You evaluated expressions like \( f(g(x)) \) using this function. For \( g(x) = x^2 \), this means that when you introduce an input \( x \), the output is that input multiplied by itself.This concept becomes particularly interesting when combined with exponential functions to form composite functions, like \( f(g(x)) = e^{x^2} \), where you first square \( x \) and then calculate the exponential of that result. This shows how combining simple functions can create more complex expressions.