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16 Cards in this Set
- Front
- Back
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Describe how the graph of f(x)=9x-2 would change if the y-intercept is changed to +8.
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The slope of the line would remain the same, but the graph would be shifted up 10 units on the y-axis.
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Determine whether the function is odd, even, or neither. f(x)=3^x+3^-x
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even
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Determine whether the function is odd, even, or neither. h(x)=x2+4x-2
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neither
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Determin whether the function is odd, even, or neither. g(x)=x^3-4x
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odd
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Describe how the graph of f(x)= 4x+3 would change if the slope were changed to -4.
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The graph would have the same y-intercept, but would travel from Quadrant II to Quadrant IV rather than Quadrant III to Quadrant I.
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Describe how the graph of f(x) =(2/3)x-5 would change if the entire function were multiplied by 3.
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The slope of the graph would be much greater and the y-intercept would be 10 units lower.
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Let f be the function that assigns to a temperature in degrees Celsius its equivalent in degrees Fahrenheit. The freezing point of water in degrees Celsius is 0 while in degrees Fahrenheit it is 32. The boiling point of water is 100 degrees Celsius and 212 degrees Fahrenheit. Given that the function f is linear, find an equation for f.
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since f is a linear function of x, the temperature in degrees Celsius, we can write f(x)=ax+b. We are given that 0 degrees Celsius converts to 32 degrees Fahrenheit so 32=f(0)=b. In order to find the slope, a, we can use the second piece of data given, namely that 100 degrees Celsius converts to 212 degrees Fahrenheit 212=f(100)=a(100)+32 a=9/5, so f(x)=9/5x+32
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Let f be the function that assigns to a temperature in degrees Celsius its equivalent in degrees Fahrenheit. Find the inverse of function f and explain its meaning in terms of temperature conversions.
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f(x)=9/5x+32 so it's inverse must be subtracting 32 from its input and then dividing by 9/5, whcih can be shown as g(x)=5/9(x-32)
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Is there a temperature which is the same in degrees Celsius and in degrees Fahrenheit? Explain how you know.
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If there is a temperature x which is the same in degrees C and F, then we would have f(x)=x. If you solve 9/5x+32=x, you find that x=-40. So -40 degrees is the temperature which registers the same in degrees C and F.
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Let f be the function defined by f(x)=10^x and g be the function defined by g(x)=log10(x). Sketch the graph of y=f(g(x)). Explain your reasoning.
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We are trying to sketch f(g(x))=10^logx. The graph should look like y=x with a limited domain because the domain is x>0
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Let f be the function defined by f(x)=10^x and g be the function defined by g(x)=log10(x). Sketch the graph of g(f(x)). Explain your reasoning.
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We are trying to sketch g(f(x))= log10^x because the domain of f(x)=10^x is any real number and its range is all positive real numbers. So, all values of f(x) are in the domain g(x)=logb(x), so there is no restriction on the domain and the graph should look like y=x
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Let f and g be any two inverse functions. For which values of x does f(g(x))=x? For which values of x does g(f(x))=x?
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If f and g are inverse functions, then g(f(x))=x for all values of x in the domain of f and f(g(x))=x for all values of x in the domain of g.
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How do you know if a function is invertible?
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A function is invertible if every output corresponds to one and only one input.
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Recall that logb(x) is by definition the exponent which b must be raised in order to yield x (b>0). Use this definition to compute log₂(2⁵).
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log₂(2⁵)=5 because 5 is the exponent 2 must be raised to in order to yield 2⁵.
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Recall that logb(x) is by definition the exponent which b must be raised in order to yield x (b>0). Use this definition to compute log₁₀(0.001).
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log₁₀(0.001)=-3 because -3 is the exponent 10 is raised to in order to yield 0.001.
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Recall that logb(x) is by definition the exponent which b must be raised in order to yield x (b>0). Explain why logb(b^y) =y where b>0.
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logb(b^y)=y (if b>0) because y is the exponent b must be raised to in order to yield b^y. In other words, when we are finding a logarithm, we are asking "what is the exponent on b in the expression b^y? under this verbal description, the answer is clearly y.
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