Textbook Question
Composition of Functions
Evaluate each expression using the functions
f(x) = 2 − x, g(x) = { −x, −2 ≤ x < 0
x − 1, 0 ≤ x ≤ 2
c. g(g(−1))
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0. Functions
Combining Functions
2:24 minutesProblem 1.2.21
Textbook Question
Composition of Functions
A balloon’s volume V is given by V = s² + 2s + 3 cm³, where s is the ambient temperature in °C. The ambient temperature s at time t minutes is given by s = 2t − 3 °C. Write the balloon’s volume V as a function of time t.
Verified step by step guidance1
Identify the given functions: The volume of the balloon is given by \( V = s^2 + 2s + 3 \) and the ambient temperature is given by \( s = 2t - 3 \).
Recognize that you need to find the composition of functions, specifically \( V(s(t)) \), which means substituting the expression for \( s \) into the volume function \( V \).
Substitute \( s = 2t - 3 \) into the volume function \( V = s^2 + 2s + 3 \). This gives \( V = (2t - 3)^2 + 2(2t - 3) + 3 \).
Expand the expression \( (2t - 3)^2 \) using the formula \( (a - b)^2 = a^2 - 2ab + b^2 \). This results in \( 4t^2 - 12t + 9 \).
Combine all terms: \( V = 4t^2 - 12t + 9 + 4t - 6 + 3 \). Simplify the expression to write the volume \( V \) as a function of time \( t \).
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Video duration:
2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Composition of Functions
Composition of functions involves combining two functions to create a new function. In this case, we have two functions: the volume V as a function of temperature s, and the temperature s as a function of time t. To find V as a function of t, we substitute the expression for s into the equation for V, effectively creating a new function that relates volume directly to time.
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Quadratic Functions
A quadratic function is a polynomial function of degree two, typically expressed in the form V = ax² + bx + c. In the given volume equation V = s² + 2s + 3, the variable s is raised to the second power, indicating that the volume changes in a parabolic manner with respect to temperature. Understanding the properties of quadratic functions, such as their vertex and axis of symmetry, is essential for analyzing their behavior.
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Substitution Method
The substitution method is a technique used in algebra and calculus to simplify expressions or solve equations by replacing a variable with another expression. In this problem, we substitute the expression for s (temperature) into the volume equation V. This method allows us to express V solely in terms of t (time), facilitating the analysis of how the balloon's volume changes over time.
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