Please give an example of another function whose inverse is only defined if we restrict the domain of the original function.
2 answers:
Answer:
f(x) = x^2
Step-by-step explanation:
The square root function is defined to have a non-negative range only. That corresponds to restricting the domain of f(x) = x^2 to positive values of x.
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The attached graph shows the domain-restricted f(x)=x² in solid red and the corresponding f⁻¹(x) = √x in solid blue. The other halves of those curves are shown as dotted lines (and are inverse functions of each other, too). The dashed orange line is the line of reflection between a function and its inverse.
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Answer:
- Part (a) Vertex = (-3, -1)
- Part (b) Vertex = (4, 5)
- Part (c) function 2; max value = 5
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Explanations:
Part (a)
In this case, the vertex is the highest point, so the vertex of function 1 is (-3, -1)
If the parabola was flipped so that it opened upward, instead of downward, then the vertex would be the lowest point.
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Part (b)
We could graph this function to see where the highest or lowest point is, but there's another way we could find the vertex using algebra.
The given equation is in the form y = ax^2 + bx + c where in this case
- a = -2
- b = 16
- c = -27
The a and b values are plugged into the formula below to get
h = -b/(2a)
h = -16/(2*(-2))
h = -16/(-4)
h = 4
This is the x coordinate the vertex since the vertex is (h,k)
To find k, we plug that value of h into the original function to find its corresponding y value.
y = -2x^2 + 16x - 27
y = -2(4)^2 + 16(4) - 27
y = -2(16) + 16(4) - 27
y = -32 + 64 - 27
y = 5
We found that h = 4 leads to k = 5. The vertex is (h,k) = (4, 5)
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Part (c)
We'll compare the y values of each vertex
- Function 1 has a vertex of (-3, 1)
- Function 2 has a vertex of (4, 5)
The vertex with the larger y value is the winner, so that would be function 2. This parabola reaches a higher peak compared to the other curve.
The max value here is y = 5. In other words, it's the maximum output the function can produce.
If you graphed functions 1 and 2 together on the same xy grid (see below), you'll see that the curve for function 2 reaches a higher peak.
