I need to find the intercepts for:
f(x)=e^(x^2-9x+20)-1
I assume you mean x-intercepts.
Since e^0 = 1, you just need to solve
x^2-9x+20 = 0
Back to Algebra I, now ...
for the y-intercept, let x = 0
f(0) = e^(0-0+20) - 1
= e^20 - 1
that is really big, appr 485,165,194
for the x-intercept, let y = 0
e^(x^2-9x+20)-1 = 0
e^(x^2-9x+20) = 1
e^(x^2-9x+20) = e^0
so x^2 - 9x + 20 = 0
(x-5)(x-4) = 0
x = 5 or x = 4
To find the intercepts of a function, you need to determine the x-values where the function intersects the x-axis (x-intercepts) and the y-values where the function intersects the y-axis (y-intercepts).
Let's start by finding the x-intercepts. These occur when the function f(x) equals zero. In other words, we need to solve the equation:
e^(x^2 - 9x + 20) - 1 = 0
To solve this equation, we can rearrange it as:
e^(x^2 - 9x + 20) = 1
Now, take the natural logarithm of both sides of the equation:
ln(e^(x^2 - 9x + 20)) = ln(1)
Using the logarithmic property that ln(e^a) = a, we get:
x^2 - 9x + 20 = 0
This is a quadratic equation, which can be factored or solved using the quadratic formula. In this case, the equation can be factored as:
(x - 4)(x - 5) = 0
Setting each factor equal to zero, we have:
x - 4 = 0 or x - 5 = 0
Solving these equations, we find:
x = 4 or x = 5
So, the x-intercepts of the function f(x) are x = 4 and x = 5.
Next, let's find the y-intercept. The y-intercept occurs when x is equal to zero. We can substitute x = 0 into the function f(x) to find the corresponding y-value:
f(0) = e^(0^2 - 9(0) + 20) - 1
= e^(20) - 1
To find the numerical value, you need to use a calculator or a computer algebra system to compute e^(20). The result is approximately 485165.19. Subtracting 1, we get:
f(0) ≈ 485165.19 - 1
≈ 485165.18
So, the y-intercept of the function f(x) is approximately y = 485165.18.
To summarize, the intercepts of the function f(x)=e^(x^2-9x+20)-1 are x = 4 and x = 5 for the x-intercepts, and approximately y = 485165.18 for the y-intercept.