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8. Given f(x)=x+3 and g(x)=4 – x², find g(f(x)) and f(g(x)). 9. Given f(x)=x²+4x-9 and g(x)=-4x² – 5x, find (f+g)(x) and (f-g)(x).

8. Given f(x)=x+3 and g(x)=4 – x², find g(f(x)) and f(g(x)). 9. Given f(x)=x²+4x-9 and g(x)=-4x² – 5x, find (f+g)(x) and (f-g)(x).

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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### Composite and Combined Functions #### Problem 8 Given \( f(x) = x + 3 \) and \( g(x) = 4 - x^2 \), find \( g(f(x)) \) and \( f(g(x)) \). **Solution:** - To find \( g(f(x)) \): 1. Substitute \( f(x) \) into \( g \): \[ g(f(x)) = g(x + 3) \] 2. Since \( g(x) = 4 - x^2 \), replace \( x \) with \( x+3 \): \[ g(x+3) = 4 - (x+3)^2 \] 3. Simplify: \[ g(x+3) = 4 - (x^2 + 6x + 9) \] \[ g(x+3) = 4 - x^2 - 6x - 9 \] \[ g(x+3) = -x^2 - 6x - 5 \] - To find \( f(g(x)) \): 1. Substitute \( g(x) \) into \( f \): \[ f(g(x)) = f(4 - x^2) \] 2. Since \( f(x) = x + 3 \), replace \( x \) with \( 4 - x^2 \): \[ f(4 - x^2) = (4 - x^2) + 3 \] \[ f(4 - x^2) = 7 - x^2 \] #### Problem 9 Given \( f(x) = x^2 + 4x - 9 \) and \( g(x) = -4x^2 - 5x \), find \( (f+g)(x) \) and \( (f-g)(x) \). **Solution:** - To find \( (f + g)(x) \): 1. Combine \( f(x) \) and \( g(x) \): \[ (f+g)(x) = f(x) + g(x) \] \[ (f+g)(x) = (x^2 + 4x - 9) + (-4x^2 - 5x) \]
Transcribed Image Text:### Composite and Combined Functions #### Problem 8 Given \( f(x) = x + 3 \) and \( g(x) = 4 - x^2 \), find \( g(f(x)) \) and \( f(g(x)) \). **Solution:** - To find \( g(f(x)) \): 1. Substitute \( f(x) \) into \( g \): \[ g(f(x)) = g(x + 3) \] 2. Since \( g(x) = 4 - x^2 \), replace \( x \) with \( x+3 \): \[ g(x+3) = 4 - (x+3)^2 \] 3. Simplify: \[ g(x+3) = 4 - (x^2 + 6x + 9) \] \[ g(x+3) = 4 - x^2 - 6x - 9 \] \[ g(x+3) = -x^2 - 6x - 5 \] - To find \( f(g(x)) \): 1. Substitute \( g(x) \) into \( f \): \[ f(g(x)) = f(4 - x^2) \] 2. Since \( f(x) = x + 3 \), replace \( x \) with \( 4 - x^2 \): \[ f(4 - x^2) = (4 - x^2) + 3 \] \[ f(4 - x^2) = 7 - x^2 \] #### Problem 9 Given \( f(x) = x^2 + 4x - 9 \) and \( g(x) = -4x^2 - 5x \), find \( (f+g)(x) \) and \( (f-g)(x) \). **Solution:** - To find \( (f + g)(x) \): 1. Combine \( f(x) \) and \( g(x) \): \[ (f+g)(x) = f(x) + g(x) \] \[ (f+g)(x) = (x^2 + 4x - 9) + (-4x^2 - 5x) \]
## Mathematics Problems on Functions ### Problem 6: **Find all the “zeros” of the function \( f(x) = x^3 - 5x^2 + 4x - 20 \) if you know \( x = 5 \) is a root of \( f \):** To solve this, perform polynomial division or synthetic division to factorize the polynomial \( f(x) \). Use the root \( x = 5 \) to find the remaining factors and then solve for the other zeros. ### Problem 7: **Given the function \( f(x) = x^2 \):** a) **Find the function \( g(x) \) performing on \( f(x) \) a reflection in the X-axis followed by a downward shift of 5 units:** - **Reflection in the X-axis:** \( g_1(x) = -f(x) = -x^2 \) - **Downward shift of 5 units:** \( g(x) = g_1(x) - 5 = -x^2 - 5 \) b) **Find the function \( h(x) \) performing on \( f(x) \) a horizontal shift of three units to the right followed by a reflection in the X-axis:** - **Horizontal shift of 3 units to the right:** \( h_1(x) = f(x-3) = (x-3)^2 \) - **Reflection in the X-axis:** \( h(x) = -h_1(x) = -(x-3)^2 \) These transformations provide new functions based on the given quadratic function \( f(x) = x^2 \).
Transcribed Image Text:## Mathematics Problems on Functions ### Problem 6: **Find all the “zeros” of the function \( f(x) = x^3 - 5x^2 + 4x - 20 \) if you know \( x = 5 \) is a root of \( f \):** To solve this, perform polynomial division or synthetic division to factorize the polynomial \( f(x) \). Use the root \( x = 5 \) to find the remaining factors and then solve for the other zeros. ### Problem 7: **Given the function \( f(x) = x^2 \):** a) **Find the function \( g(x) \) performing on \( f(x) \) a reflection in the X-axis followed by a downward shift of 5 units:** - **Reflection in the X-axis:** \( g_1(x) = -f(x) = -x^2 \) - **Downward shift of 5 units:** \( g(x) = g_1(x) - 5 = -x^2 - 5 \) b) **Find the function \( h(x) \) performing on \( f(x) \) a horizontal shift of three units to the right followed by a reflection in the X-axis:** - **Horizontal shift of 3 units to the right:** \( h_1(x) = f(x-3) = (x-3)^2 \) - **Reflection in the X-axis:** \( h(x) = -h_1(x) = -(x-3)^2 \) These transformations provide new functions based on the given quadratic function \( f(x) = x^2 \).
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