1.) Use the graph of f to describe the transformation that yields the graph of g.
f(x)=10^x
g(x)= 10^-x+3
a.) Reflect the graph of f in the y-axis and shift three units to the left.
b.) Reflect the graph of f in the y-axis and shift three units to the right.
c.) Reflect the graph of f in the x-axis and shift three units to the right.
d.) Reflect the graph of f in the y-axis and shift three units up
e.) Reflect the graph of f in the x-axis and shift three units to the left.
f.) Reflect the graph of f in the x-axis and shift three units up.
2.) Use the One-to-One Property to solve the equation for x.
e^x^2+6=e^5x
#1 d
e^(x^2+6) = e^5x
x^2+6 = 5x
x^2-5x+6 = 0
x = 2,3
1.) The correct answer is d.) Reflect the graph of f in the y-axis and shift three units up.
Explanation:
To understand the transformation that yields the graph of g, we should analyze the function g(x).
The function g(x) = 10^-x+3 can be rewritten as g(x) = 10^-x * 10^3.
The graph of f(x) = 10^x is an exponential function that starts at the point (0,1) and increases rapidly as x increases.
Reflecting the graph of f in the y-axis means changing the sign of x. So, we have 10^-(-x) * 10^3 = g(x).
Simplifying further, we get 10^x * 10^3 = g(x).
The addition of 3 in the exponent of 10 means shifting the graph three units up.
Therefore, the correct answer is d.) Reflect the graph of f in the y-axis and shift three units up.
2.) To solve the equation e^x^2+6 = e^5x using the One-to-One Property, we will set the exponents equal to each other:
x^2 + 6 = 5x
Now, we rearrange the equation to get it in standard quadratic form:
x^2 - 5x + 6 = 0
We factor the quadratic:
(x - 2)(x - 3) = 0
To solve for x, we set each factor equal to zero:
x - 2 = 0 or x - 3 = 0
Solving for x gives us two possible solutions:
x = 2 or x = 3
Therefore, the solutions for the equation e^x^2+6 = e^5x using the One-to-One Property are x = 2 and x = 3.
1.) To determine the transformation that yields the graph of g, we need to analyze the equation g(x) and compare it to the equation of f(x).
f(x) = 10^x represents an exponential function where the base is 10 and x is the exponent.
Now let's analyze g(x) = 10^(-x) + 3.
The negative exponent (-x) in g(x) reflects the graph of f(x) across the y-axis.
Adding 3 to the function shifts the graph three units upward.
Therefore, the correct answer is d.) Reflect the graph of f in the y-axis and shift three units up.
2.) The One-to-One Property states that if two exponential expressions with the same base are equal, then their exponents must be equal.
Using the given equation e^x^2 + 6 = e^5x, we can solve it using the One-to-One Property.
First, we subtract e^5x from both sides to isolate the exponential expression with x^2:
e^x^2 - e^5x + 6 = 0
Now, let's rewrite the expression using a substitution.
Let y = e^x
Substituting y in the equation, we have:
y^2 - e^5 * y + 6 = 0
Now, we have a quadratic equation which we can solve using factoring, completing the square, or the quadratic formula.
In this case, the equation can be factored as:
(y - 2)(y - 3) = 0
Setting each factor equal to 0, we have two solutions:
y - 2 = 0, or y - 3 = 0
Solving for y, we find:
y = 2, or y = 3
Remember, we substituted y = e^x, so now we substitute back to solve for x:
For y = 2:
e^x = 2
Taking the natural logarithm (ln) of both sides, we get:
x = ln(2)
For y = 3:
e^x = 3
Taking the natural logarithm (ln) of both sides, we get:
x = ln(3)
Therefore, the solutions for x are x = ln(2) and x = ln(3).