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Magazine Chapter 1: Problem 5
For the functions \(f\) and \(g\), find (a) \(f(g(1))\) (b) \(g(f(1))\) (c) \(f(g(x))\) (d) \(g(f(x))\) (e) \(f(t) g(t)\) $$f(x)=e^{x}, g(x)=x^{2}$$
Short Answer
Expert verified
(a) \(e\); (b) \(e^2\); (c) \(e^{x^2}\); (d) \(e^{2x}\); (e) \(t^2 e^t\).
Step by step solution
01
Evaluate g(1)
First, we need to find the value of the function \( g \) at \( x=1 \). Since \( g(x) = x^2 \), we substitute 1 into the expression, so \( g(1) = 1^2 = 1 \).
02
Evaluate f(g(1))
Now we know \( g(1) = 1 \), we find \( f(g(1)) = f(1) \). Using the function \( f(x) = e^x \), substitute \( x = 1 \) to get \( f(1) = e \). So, \( f(g(1)) = e \).
03
Evaluate f(1)
Next, we find the value of \( f(1) \) for part b. Using \( f(x) = e^x \), substitute \( x = 1 \) to get \( f(1) = e \).
04
Evaluate g(f(1))
Using the value from Step 3, where \( f(1) = e \), we now find \( g(f(1)) = g(e) \). Since \( g(x) = x^2 \), substitute \( x = e \) to get \( g(e) = e^2 \).
05
Find f(g(x))
To express \( f(g(x)) \), substitute \( g(x) = x^2 \) into \( f(x) = e^x \). We get \( f(g(x)) = f(x^2) = e^{x^2} \).
06
Find g(f(x))
To find \( g(f(x)) \), substitute \( f(x) = e^x \) into \( g(x) = x^2 \). We find \( g(f(x)) = g(e^x) = (e^x)^2 = e^{2x} \).
07
Find f(t) g(t)
Finally, to determine \( f(t)g(t) \), use the functions \( f(t) = e^t \) and \( g(t) = t^2 \). Multiply them together to get \( f(t)g(t) = e^t \, t^2 = t^2 e^t \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Exponential Function
An exponential function is a mathematical function of the form \( f(x) = e^x \), where \( e \) is the base of the natural logarithm. The value of \( e \) is approximately equal to 2.71828. Exponential functions are widely used in various fields such as physics, biology, and finance owing to their ability to model growth and decay processes.
Exponential functions have unique features that make them special:
Exponential functions have unique features that make them special:
- Their rate of growth increases rapidly as the input value \( x \) increases.
- They have a constant base \( e \), which is an irrational number.
- They are always positive for real number inputs, producing only positive output values.
Quadratic Function
A quadratic function is a type of polynomial function that takes the form \( g(x) = x^2 \). The graph of a quadratic function is a parabola, which can open upwards or downwards depending on the sign of the coefficient of \( x^2 \).
Quadratic functions are characterized by:
Quadratic functions are characterized by:
- A single variable raised to the second power.
- A symmetrical graph with a vertex, which is the highest or lowest point of the parabola.
- The ability to model areas associated with quadratic equations, motions, and trajectories.
Function Evaluation
Function evaluation involves finding the output of a function given a particular input. This concept is crucial when working with different types of functions, such as exponential and quadratic functions.
Here's how you evaluate functions:
Here's how you evaluate functions:
- Substitute the given input value into the function in place of the variable. For example, for \( f(x) = e^x \), finding \( f(1) \) means substituting \( x = 1 \) into the equation to get \( e^1 = e \).
- Perform the necessary mathematical operations according to the function's definition and simplify if possible.
- Be mindful of the domain of the function to ensure the input value is within its allowable range.
Multiplication of Functions
In mathematics, multiplying functions involves taking two functions and multiplying their outputs for the same independent variable. The notation \( (f \, \cdot \, g)(x) \) represents the product of functions \( f \) and \( g \), calculated as \( f(x) \, g(x) \).
Here are some key points about function multiplication:
Here are some key points about function multiplication:
- You perform multiplication by finding the product of each function's result at the same input. For instance, if \( f(t) = e^t \) and \( g(t) = t^2 \), then \( f(t)g(t) = e^t \, t^2 \).
- It is crucial to multiply the outputs rather than the expressions directly; this preserves the proper functional form.
- Multiplying functions allows you to find the combined effect of two functions acting over the same variable, such as calculating areas under curves defined by multiplying functions.
