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Questions and Answers
If f(x) = 3x + 2 and g(x) = x^2 + 1, which expression is equivalent to (f*g)(x)?
If f(x) = 3x + 2 and g(x) = x^2 + 1, which expression is equivalent to (f*g)(x)?
- 3(x^2) + 2
- 3x^2 + 2
- 2(x^2 + 1) + 3
- 3(x^2 + 1) + 2 (correct)
If u(x) = -2x^2 + 3 and v(x) = 1/x, what is the range of (u*v)(x)?
If u(x) = -2x^2 + 3 and v(x) = 1/x, what is the range of (u*v)(x)?
- (-∞, 0)
- (-∞, 3) (correct)
- (0, ∞)
- (3, ∞)
If h(x) = 6 - x, what is the value of (h*h)(10)?
If h(x) = 6 - x, what is the value of (h*h)(10)?
10
Which of the following describes the domain of (g*f)(x) if the domain of f(x) is all real values except 7 and the domain of g(x) is all real values except -3?
Which of the following describes the domain of (g*f)(x) if the domain of f(x) is all real values except 7 and the domain of g(x) is all real values except -3?
If p(x) = 2x^2 - 4x and q(x) = x - 3, what is (p*q)(x)?
If p(x) = 2x^2 - 4x and q(x) = x - 3, what is (p*q)(x)?
If h(x) = x - 7 and g(x) = x^2, which expression is equivalent to (g*h)(5)?
If h(x) = x - 7 and g(x) = x^2, which expression is equivalent to (g*h)(5)?
If f(x) = x^2 + 1 and g(x) = x - 4, which value is equivalent to (f*g)(10)?
If f(x) = x^2 + 1 and g(x) = x - 4, which value is equivalent to (f*g)(10)?
Which description best explains the domain of (g*f)(x)?
Which description best explains the domain of (g*f)(x)?
If s(x) = x - 7 and t(x) = 4x^2 - x + 3, which expression is equivalent to (t*s)(x)?
If s(x) = x - 7 and t(x) = 4x^2 - x + 3, which expression is equivalent to (t*s)(x)?
If h(x) = 5 + x and k(x) = 1/x, which expression is equivalent to (k*h)(x)?
If h(x) = 5 + x and k(x) = 1/x, which expression is equivalent to (k*h)(x)?
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Study Notes
Composition of Functions Study Notes
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Composition of Functions: A function composed of two functions combines their outputs. Denoted as (f * g)(x) or (g * f)(x), it evaluates one function using the output of another.
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Example of Composition: For f(x) = 3x + 2 and g(x) = x² + 1, (f * g)(x) simplifies to 3(x² + 1) + 2, illustrating how to apply functions sequentially.
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Range of Function Composition: When u(x) = -2x² + 3 and v(x) = 1/x, the range of (u * v)(x) is (-∞, 3), indicating limits on possible output values based on input values.
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Evaluating Function Values: For h(x) = 6 - x, calculating (h * h)(10) results in 10, showing the function can be recursively applied.
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Domain of Composed Functions: If the domain of f(x) excludes 7 and g(x) excludes -3, the domain of (g * f)(x) excludes both x = 7 and values for which f(x) = -3, combining restrictions from both functions.
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Multiplication of Functions: Given p(x) = 2x² - 4x and q(x) = x - 3, (p * q)(x) results in 2x² - 16x + 30, showing how to expand function products.
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Calculating Composite Function Values: For h(x) = x - 7 and g(x) = x², (g * h)(5) leads to expression (5 - 7)², demonstrating the process for specific input evaluations.
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Composite Function Outputs: For f(x) = x² + 1 and g(x) = x - 4, the evaluation of (f * g)(10) gives 37, highlighting input-specific outputs based on function definitions.
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Understanding Domains in Composition: The domain of (g * f)(x) is defined by the inputs of f(x) for which g(f(x)) remains valid.
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Expanded Form in Composition: When s(x) = x - 7 and t(x) = 4x² - x + 3, the equivalent expression for (t * s)(x) is 4(x - 7)² - (x - 7) + 3, displaying function expression manipulation.
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Reciprocal Function Composition: With h(x) = 5 + x and k(x) = 1/x, the composition (k * h)(x) translates to 1/(5 + x), reflecting the application of reciprocal functions.
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